Kinetic Parameter Estimation of Gelatin Waste by Thermogravimetry

Chemical Engineering Journal125(2006)111–117Kinetic modelling of the adsorption of nitratesby ion exchange resinM.Chabani a,A.Amrane b,∗,A.Bensmaili aa Facult´e de G´e nie des Proc´e d´e s et G´e nie M´e canique,U.S.T.H.B.BP32,El Allia,Bab ezzouar,Algeriab Equipe Chimie et Ing´e nierie des Proc´e d´e s-ENSCR/Universit´e de rennes1,UMR CNRS6226“Sciences chimiques de Rennes”,ENSCR,Campus de Beaulieu,av.du G´e n´e ral Leclerc,35700Rennes,FranceReceived21April2006;received in revised form17July2006;accepted6August2006AbstractThe capacity of ion exchange resins,Amberlite IRA400,for removal of nitrates from aqueous solution was investigated under different initial concentrations.The suitability of the Freundlich,Langmuir and Dubinin-Radushkevich adsorption models to the equilibrium data was investigated.The equilibrium data obtained in this study were found to follow Freundlich adsorption isotherm.The maximum sorption capacity was769.2mg/g at25◦C.Reversible-first-order,intraparticle diffusion,film diffusion and Bangham models were used tofit the experimental data. The adsorption of nitrates on Amberlite IRA400resin follows reversible-first-order kinetics.The overall rate constants were estimated for different initial concentrations.Results of the intra-particle diffusion and thefilm diffusion models show that thefilm diffusion was the main rate-limiting step.The low correlation of data to the Bangham’s equation also confirms that diffusion of nitrates into pores of the resin was not the only rate-controlling step.The thermodynamic constants of adsorption phenomena, H◦and S◦were found to be−26.122kJ/mol and−68.76J/mol in the range298–318K and+19.205kJ/mol and+68.76J/mol in the range318–343K,respectively.The negative values of the Gibbs free energy ( G)demonstrate the spontaneous nature of adsorption of nitrates onto Amberlite IRA400.©2006Elsevier B.V.All rights reserved.Keywords:Adsorption;Resin;Nitrates;Kinetics1.IntroductionSeveral nitrogenous compounds,including ammonia,nitrite and nitrate are frequently present in drinking water[1].Nitrates can cause several environmental problems.Nitrates and phos-phates can stimule eutrophication where pollution is caused in waterways by heavy algal growth,as they are both rate-limiting nutrients for process.A high concentration of nitrate-nitrogen in drinking water leads to the production of nitrosamine,which is related to cancer,and increases the risks of diseases such as methemoglobinemia in new-born infants[2,3].Several methods that serve to reduce nitrates in drinking water have been developed.The use of biological reactor seems to be the most promising technique in the treatment of high nitrate concentrations.However,maintaining biological processes at their optimum conditions is difficult,and the problems of con-∗Corresponding author.Tel.:+33223235755;fax:+33223238120.E-mail address:abdeltif.amrane@univ-rennes1.fr(A.Amrane).tamination by dead-bacteria have to be solved to make such processes satisfactory for a safety use in drinking water treat-ment.Adsorption is a useful process for in situ treatment of underground and surface water,primarily due to it’s easy of use [3].Adsorbent resins are considered the most promising owing to their chemical stability and ability to control surface chemistry [4].The characteristics of adsorption behaviour are generally inferred in terms of both adsorption kinetics and equilibrium isotherms.They are also important tools to understand the adsorption mechanism,viz.the theoretical evaluation and inter-pretation of thermodynamic parameters[5,6].The objective of this study was to investigate equilibrium and kinetic parameters for the removal of nitrates from aqueous solutions by adsorption onto an ionized adsorbent,Amberlite IRA400.The Langmuir,Freundlich and Dubinin-Radushkevich (D-R)equations were used tofit the equilibrium isotherms.Ther-modynamic parameters were also evaluated through adsorption measurements.1385-8947/$–see front matter©2006Elsevier B.V.All rights reserved. doi:10.1016/j.cej.2006.08.014112M.Chabani et al./Chemical Engineering Journal125(2006)111–117Nomenclatureb Langmuir constant(cm3/mg)C0initial concentration in solution(mg/l)C e equilibrium concentration(mg/l)C t concentration at time t(mg/l)D p pore diffusion coefficient(cm2/s)D ffilm diffusion coefficient(cm2/s)E free energy of adsorption(kJ/mol)G Gibbs free energy(kJ/mol)H◦enthalpy of adsorption(kJ/mol)k overall rate constant of adsorption(min−1)k1forward rate constant(min−1)k2backward rate constant(min−1)k d distribution constant(l/g)k f Freundlich’s constant related to the sorptioncapacityk i intraparticle diffusion parameter(mg/g min0.5)m mass of adsorbent(g)n Freundlich’s constant related to the sorptionintensity of a sorbentq the amount of nitrates adsorbed or transferred(mg/l)q o maximum adsorption capacity(mg/g)q e adsorption capacity in equilibrium(mg/g)q t amount of adsorption at time t(mg/g)R ideal gas constant(8.314J/mol K)R p radius of the adsorbent(cm)S◦entropy of adsorption(J/mol K)t time(min)t1/2time for half adsorption(min)T temperature(K)v volume of solution(l)Greek letterεfilm thickness(cm)2.Materials2.1.Pre-treatment of resinBefore use,the resin was washed in distilled water to remove the adhering dirt and then dried at50◦C.After drying,the resin was screened to obtain a particle size range of0.3–0.8mm.2.2.The resin characteristicsThe main characteristics of the Amberlite IRA400(Merck, Darmstadt,Germany),a macroporous anion exchange resin,are given in Table1.2.3.Nitrate solutionsThe stock solution of NO3−used in this study was prepared by dissolving an accurate quantity of KNO3in distilled water.Table1Characteristics of Amberlite IRA400ion exchange resinPolymer matrix Polystyrene DVBFunctional group–N+R3Ionic form Cl−Exchange capacity 2.6–3eq kg−1of dry massAppearance Yellow to golden spherical beads,translucent Effective size0.3–0.8mmWater retention42–48%Visual density in wet state0.66–0.73g/mlTrue density in wet state 1.07–1.10g/mlA range of dilutions,1–18mg/l,was prepared from the stock solution.2.4.Sorption experimentsThe pH of the aqueous solutions of NO3−was approximately 6.8and did not varied significantly with the dilution.The effect of temperature on the sorption rate of nitrates on Amberlite IRA400was investigated by equilibrating the sorption mix-ture(800ml)containing dried resin(2g)and nitrates(15mg/l) in a temperature range298–343K.The solutions were placed in flasks and stirred for3h.Experiments were mainly carried out without initial adjustment of the pH.Preliminary tests showed that the adsorption was complete after1h.The concentration of residual nitrate ions was determined spectrophotometrically according to Rodier protocol[7].The sorption capacity at time t,q t(mg/g)was obtained as follows: q t=(C0−C t)vm(1) where C0and C t(mg/l)were the liquid-phase concentrations of solutes at the initial and a given time t,respectively,v(l)the volume of solution and m the mass resin(g).The amount of adsorption at equilibrium,q e was given by: q e=(C0−C e)vm(2) C e(mg/l)was the concentration of nitrates at equilibrium.The distribution constant k d was calculated using the follow-ing equation:k d=Amount of nitrates in adsorbentAmount of nitrates in solution×vm.(3) 3.Results and discussion3.1.Adsorption isothermsSorption equilibrium is usually described by an isotherm equation whose parameters express the surface properties and affinity of the sorbent,at afixed temperature and pH[8].An adsorption isotherm describes the relationship between the amount of adsorbate adsorbed on the adsorbent and the con-centration of dissolved adsorbate in the liquid at equilibrium. Equations often used to describe the experimental isotherm data are those developed by Freundlich[9],by Langmuir[10]and by Dubinin-Radushkevich[11].The Freundlich and LangmuirM.Chabani et al./Chemical Engineering Journal 125(2006)111–117113isotherms are the most commonly used to describe the adsorp-tion characteristics of adsorbent used in water and wastewater.3.1.1.Freundlich isotherm (1906)The empirical model can be applied to non-ideal sorption on heterogeneous surfaces as well as multilayer sorption and is expresses by the following equation:q e =k f C 1/ne(4)n and k f are the Freundlich constants.The fit of data to Freundlich isotherm indicates the hetero-geneity of the sorbent surface.The magnitude of the exponent 1/n gives an indication of the adequacy and capacity of the adsor-bent/adsorbate system [12].In most cases,an exponent between 1and 10shows beneficial adsorption.The linear plot of ln q e versus ln C e (Fig.1)shows that the adsorption obeys to the Fre-undlich model.k f and n were determined from the Freundlich plot and found to be 0.982and 1.074(Fig.1),respectively.n values above 1indicate favourable adsorption.ngmuir isotherm (1916)The Langmuir model is probably the best known and most widely applied sorption isotherm.It may be represented as fol-lows:q e =q o bC e1+bC e (5)The Langmuir constants q o and b are related to the adsorption capacity and the energy of adsorption,respectively.The linear plot of 1/q e versus 1/C e shows that the adsorption obeys to the Langmuir model (Fig.2),and gave the following values for q o and b ,769.2mg/g and 1.02cm 3/mg,respectively.3.1.3.Dubinin-Radushkevich (D-R)isothermRadushkevich [13]and Dubinin [14]have reported that the characteristic sorption curve is related to the porous structure of the sorbent.The sorption data were applied to the D-R model in order to distinguish between physical and chemical adsorption [15].The D-R equation was given by:ln q e =ln q o −βε2(6)Fig.1.Linearized Freundlich isotherm (Eq.(4))for nitrates adsorption by Amberlite IRA 400;T =25◦C,pH 6.8,adsorbent dose 1.25g/l.Fig.2.Linearized Langmuir isotherm (Eq.(5))for nitrates adsorption by Amber-lite IRA 400;T =25◦C,pH 6.8,adsorbent dose 1.25g/l.where βwas the activity coefficient related to mean sorption energy and εthe Polanyi potential given by:ε=RT ln1+1e (7)where R was the gas constant (kJ/mol K)and T the temperature (K).The slope of the plot ln q e versus εgives β(mol 2/J 2)and the ordinate intercept yields the sorption capacity q o (mg/g).The mean sorption energy (E )is given by E =1/√−2β;for a magnitude of E between 8and 16kJ/mol,the adsorption process follows chemical ion-exchange [6,16],while values of E below 8kJ/mol characterize a physical adsorption process [17].The plot of ln q e against ε2for nitrate ions sorption on resin is shown in Fig.3.The E value was 40.8J/mol,which corresponds to a physical adsorption.The linear correlation coefficients for Freundlich,Langmuir and D-R are shown in Table 2,and are always greater than 0.980.The Freundlich equation represents a better fit of experimental data than Langmuir and D-Requations.Fig.3.Radushkevich-Dubinin isotherm (Eq.(6))for nitrates adsorption by Amberlite IRA 400;T =25◦C,pH 6.8,adsorbent dose 1.25g/l.114M.Chabani et al./Chemical Engineering Journal 125(2006)111–117Table 2Isotherm parameters collected for removal of nitrates by Amberlite IRA 400Freundlich constants k f 0.982n 1.074R 20.9951Langmuir constants q o (mg/g)769.2b (cm 3/mg) 1.02R 20.9933D-R constants q o (mg/g)76.5E (J/mol)40.8R 20.9873.2.Adsorption kinetics3.2.1.Effect of initial concentrationsPredicting the adsorption rate,in addition to the adsorbate residence time and the reactor dimensions,controlled by the system’s kinetics,are probably the most important factors in adsorption system design [18].Preliminary experiments showed high initial rates of adsorp-tion of nitrates followed by lower rates near equilibrium.Kinet-ics of nitrates removal at 298K showed high rates during the initial 15min,and decreased thereafter until nearly constant val-ues after 40min of adsorption (Fig.4).In batch adsorption processes the adsorbate molecules diffuse in porous adsorbent and the rate process usually depends upon t 1/2rather than the contact time,t [19,20]:q t =k i t 0.5(8)The plot of q t ,the amount of adsorbate adsorbed per unit weight of adsorbent versus square root of time has been com-monly used to describe an adsorption process controlled by diffusion in the adsorbent particle and consecutive diffusion in the bulk of the solution [21].Therefore,as can be observed in Fig.5,adsorption was linear and charaterized by extremelyfastFig.4.Time-courses of nitrates adsorption for different initial concentrations of nitricnitrogen.Fig.5.Effect of t 1/2on adsorption of nitrates by Amberlite IRA 400(Eq.(8)),adsorbent dose 1.25g/l,pH 6.8.uptake.During this phase,nitrates were adsorbed within a t 1/2value of about 10min;this behaviour can be attributed to the rapid use of the most readily available adsorbing sites on the adsorbent surface.After this phase,adsorption became negligi-ble.It may be attributed to a very slow diffusion of adsorbed nitrates from the film surface into the micropores,which are the less accessible sites of adsorption [20,22,23].Fig.5shows two consecutive linear steps during the adsorption of nitrates on Amberlite IRA 400,and the deviation from the straight line indicates that the pore diffusion is not the only rate-limiting step.The slope of the linear portion from Fig.5was used to derive values for the rate parameter,k i ,for the intra-particle diffusion (Eq.(8)).The results show that intraparticle diffusion model was valid for the considered system.Kinetic parameters and correlation coefficients obtained by the intra-particle model are given in Table 3.The values of k i increased for increasing initial concentrations.Kinetic data can further be used to check whether pore-diffusion was the only rate-controlling step or not in the adsorp-tion system using Bangham’s equation [24–26]:log log C 0C 0−qm =log k B m2.303v+a log t (9)As observed,the model did not match experimental data(Fig.6),showing that the diffusion in the pores of the sorbent wasTable 3Intraparticle diffusion (Eq.(8))kinetic parameters for adsorption of nitrates on Amberlite IRA 400resin at different concentrations Concentration of nitric nitrogen (mg/l)Intraparticle diffusion Correlation coefficient R 2Kinetic parameters,k i (mg/g min 0.5)10.8740.5120.990 1.23.50.987 2.9150.990 3.93100.9038.75140.98211.7180.96615.5M.Chabani et al./Chemical Engineering Journal125(2006)111–117115Fig.6.Fit of adsorption kinetics at different initial concentrations of nitric nitro-gen by means of the Bangham’s model(Eq.(9)).not the only rate-controlling step.The correlation coefficients given by the Bangham’s equation(Fig.6)confirmed that the model did notfit satisfactory experimental data.Similar results were previously reported for4-CP adsorption by macroreticular resins[27]and for adsorption of4-chlorophenol by XAD-4[12].3.2.2.First-order reversible modelWhen a single species is considered to adsorb on a hetero-geneous surface,the adsorption of a solute from an aqueous solution follows reversible-first-order kinetics[28,29].The het-erogeneous equilibrium between the solute in solution and the ion exchange resin may be expressed as:A k1k2B,(10)With k1the forward reaction rate and k2the backward reaction rate constant.If C0(mg/l)was the initial concentration of nitrates and q(mg/l)the amount transferred from the liquid phase to the solid phase at a given time t,then the rate was:d q d t =−d(C0−q)d t=k(C0−q)(11)where k(min−1)is the overall reaction rate constant.Since k1 (min−1)and k2(min−1)are the rate constants for the forward and the reverse processes,the rate can be expressed as:d qd t=k1(C0−q)−k2q(12)If X e(mg/l)represents the concentration of nitrates adsorbed at equilibrium,then at equilibrium:k1(C0−X e)−k2X e=0,d qd t=(k1+k2)(X e−q)(13) after integration,The above equation can be rearranged:ln(1−U t)=−(k1+k2)t=−kt(14) where U t=q/X e and k was the overall rate constant;U t,called the fractional attainment of equilibrium of nitrates,is givenby Fig.7.Fit of adsorption kinetics at different initial concentrations of nitric nitro-gen by means of the reversible-first-order model(Eq.(14)).the following expression:U t=C0−C tC0−C e(15) where C0,C t and C e were the initial nitrates concentration,its concentration at a given time t and at equilibrium,respectively. The overall constant rate k for a given concentration corre-sponded to the slope of the straight line of the plot ln(1−U t) versus t(Fig.7).The equilibrium constant and the forward and backward constant rates k1and k2were calculated using Eq.(14)and are collected in Table4.The high correlation coefficient values,if compared to that given in the available literature[30],confirm the applicability of the model.It could be seen that the forward rate constants for the removal of nitrates were much higher than the backward rate constants for the des-orption process(Table4).The constant rates depend on the adsorption capacity,the diffusion coefficient,the effective mass transfer area,the hydrodynamic of the system and other physico-chemical parameters.Ourfinding agreed with previous results [12].3.2.3.Diffusion processIn order to assess the nature of the diffusion process respon-sible for adsorption of nitrates on Amberlite IRA400,attempts were made to calculate the coefficients of the process.Iffilm dif-fusion was to be the rate-determining step in the adsorption of nitrates on the surface of the resin,the value of thefilm diffusion coefficient(D f)should be in the range10−6to10−8cm2/s.If pore diffusion was to be rate limiting,the pore diffusion coeffi-cient(D p)should be in the range10−11to10−13cm2/s[31,32]. Assuming spherical geometry for the sorbent,the overall rate constant of the process can be correlated with the pore diffu-sion coefficient and thefilm diffusion coefficient independently according to[29]:Pore diffusion coefficient:D p=0.03R2pt1/2,(16)andfilm diffusion coefficient:D f=0.23R pε1/2q(17)116M.Chabani et al./Chemical Engineering Journal125(2006)111–117 Table4Constant rates for the removal of nitrates with Amberlite IRA400resin using reversible-first-order model(Eq.(14))C0(mg/l)Correlationcoefficients,R2Overall rate constant,k=k1+k2(min−1)Forward rate constant,k1(min−1)Backward rate constant,k2(min−1)10.9540.0860.0680.018 20.9940.0890.0770.012 3.50.9780.1160.1070.009 50.9850.1410.1290.011 100.9800.1370.1280.008 140.9750.1650.1510.013 180.9920.1630.1460.017 where R p was the radius of the adsorbent(R p=0.045cm),εthefilm thickness(10−3cm)[19],q(mg/l)the amount of nitratesadsorbed and C0the initial concentration.By considering theappropriate data and the respective overall rate constants,poreandfilm diffusion coefficients for various concentrations ofnitrates were determined(Table5).It clearly appeared thatnitrates removal on Amberlite IRA400resin was controlledbyfilm diffusion since coefficient values were in the range10−6to10−8cm2/s.3.3.Thermodynamic studiesThe amounts of sorption of nitrates ions by Amberlite IRA400were measured in the range temperature298–343K.Ther-modynamic parameters were determined using the followingrelation[15,33–36]:ln k d= S◦R−H◦RT(18)where k d was the distribution coefficient, H, S,and T the enthalpy,entropy and temperature(in Kelvin),respectively,and R was the gas constant.The values of enthalpy( H)and entropy ( S)were obtained from the slope and intercept of ln(k d)versus 1/T plot.Fig.8shows two linear parts in the temperature range explored.Afirst decreasing part for a temperature range from 318to343K,and a second and increasing linear part for a tem-perature range from298to318K.Gibbs free energy( G)was calculated using the well-known equation:G= H−T S(19) The slope was positive in the temperature range298–318K and negative in a second step,from318to343K(Fig.8).The thermodynamic parameters are collected in Table6.Table5Diffusion coefficients for the removal of nitrates by Amberlite IRA400resinC0(mg/l)Pore diffusion coefficient,Eq.(16)(×10−7cm2/s)Film diffusion coefficient, Eq.(17)(×10−8cm2/s)10.670.73 20.780.92 3.50.84 1.06 50.92 1.15 10 1.01 1.29 14 1.26 1.58 18 1.692.06Fig.8.Estimation of thermodynamic parameters(Eq.(18))for the adsorption of nitrates onto Amberlite IRA400.Table6Thermodynamic parameters for the adsorption of nitrates on Amberlite IRA400 298K308K318K318K333K343K G(kJ/mol)−5.631−4.944−4.256−2.66−3.692−4.38 H◦(kJ/mol)−26.122+19.205S◦(J/mol K)−68.76+68.76For thefirst range from298to318K,the negative value of enthalpy showed that the adsorption of nitrates was exother-mic.The negative values of adsorption entropy indicated that the adsorption process was reversible and demonstrated the sponta-neous nature of adsorption(Table6).In the range318–343K,Gibbs free energy( G)decreased for increasing temperatures,indicating that the reaction was spontaneous and favoured by higher temperatures.The positive value of enthalpy indicated that the adsorption was endothermic. The positive value of adsorption entropy showed the irreversibil-ity of the adsorption process and favoured complexation and stability of sorption.The resultant effect of complex bonding and steric hindrance of the sorbed species probably increased the enthalpy and the entropy of the system.4.ConclusionsThese results show that Amberlite IRA400is an effective adsorbent for the removal of nitrates from aqueous solution.TheM.Chabani et al./Chemical Engineering Journal125(2006)111–117117equilibrium between nitrates and resin was achieved in approx-imately60min,leading to96%removal of nitrates.The corre-lation coefficient showed that Freundlich model gave a betterfit of experimental data than Langmuir and Dubinin-Radushkevich models.Adsorption kinetics was found to followfirst order reversible model expression.Thefilm diffusion was the main rate-limiting step.Temperature variations were used to evaluate enthalpy H, entropy S and Gibbs free energy G values.The negative value of G showed the spontaneous nature of adsorption.In the tem-perature range298–318K, H and S were negative,and the reaction was exothermic and reversible.From318to343K, H and S were positive,and the reaction was endothermic and irre-versible.In addition,positive S values favoured complexation and stability of sorption.References[1]N.¨Ozt¨u rk,T.Ennil Bektas,Nitrate removal from aqueous solution byadsorption onto various materials,J.Hazard.Mater.B112(2004)155–162.[2]H.Bouwer,Agricultural contamination:problems and solutions,WaterEnviron.Technol.(1989)292–297.[3]K.Mizuta,T.Matsumoto,Y.Hatate,K.Nishihara,T.Nakanishi,Removalof nitrate-nitrogen from drinking water using bamboo powder charcoal, Bioresour.Technol.95(2004)255–257.[4]N.I.Chubar,V.F.Samanidou,V.S.Kouts,G.G.Gallios,V.A.Kanibolotsky,V.V.Strelko,I.Z.Zhuravlev,Adsorption offluoride,chloride,bromide,and bromate ions on a novel ion exchanger,J.Colloid Interface Sci.291(2005) 67–74.[5]S.J.Allen,G.Mckay,J.F.Porter,Adsorption isotherm models for basicdye adsorption by peat in single and binary component systems,J.Colloid Interface Sci.280(2004)322–333.[6]A.¨Ozcan,A.S.¨Ozcan,S.Tunali,T.Akar,I.Kiran,Determination of theequilibrium,kinetic and thermodynamic parameters of adsorption of cop-per(II)ions onto seeds of Capsicum annuum,J.Hazard.Mater.B124(2005) 200–208.[7]J.Rodier,Water Analysis,seventh ed.,Dunod,Paris,1996(in French).[8]S.Sohn,D.Kim,Modelisation of Langmuir isotherm in solution systems-definition and utilization of concentration dependent factor,Chemosphere 58(2005)115–123.[9]H.M.F.Freundlich,¨Uber die adsorption in l¨o sungen,Zeitschrift f¨u rPhysikalische Chemie(Leipzig)57A(1906)385–470.[10]ngmuir,The constitution and fundamental properties of solids andliquids,J.Am.Chem.Soc.38(11)(1916)2221–2295.[11]M.M.Dubinin,E.D.Zaverina,L.V.Radushkevich,Sorbtsiyai strukturaaktivnykh uglei.I.Issledovanie adsorbtsii organicheskikh parov,Zhurnal Fizicheskoi Khimii21(11)(1947)1351–1363.[12]M.S.Bilgili,Adsorption of4-chlorophenol from aqueous solutions byXAD-4resin:isotherm,kinetic,and thermodynamic analysis,J.Hazard.Mater.137(2006)157–164.[13]L.V.Radushkevich,Potential theory of sorption and structure of carbons,Zhurnal Fizicheskoi Khimii23(1949)1410–1420.[14]M.M.Dubinin,Modern state of the theory of volumefilling of microporeadsorbents during adsorption of gases and steams on carbon adsorbents, Zhurnal Fizicheskoi Khimii39(1965)1305–1317.[15]R.Donat,A.Akdogan,E.Erdem,H.Cetisli,Themodynamics of Pb2+andNi2+adsorption onto natural bentonite from aqueous solutions,J.Colloid Interface Sci.286(2005)43–52.[16]F.Helffrich,Ion Exchange,McGraw Hill,New York,1962.[17]M.S.Onyango,Y.Kojima,O.Aoyi,E.C.Bernardo,H.Matsuda,Adsorp-tion equilibrium modelling and solution chemistry dependence offluoride removal from water by trivalent-cation exchange zeolite F-9,J.Colloid Interface Sci.279(2004)341–350.[18]Y.S.Ho,Review of second-order models for adsorption systems,J.Hazard.Mater.136(2006)681–689.[19]W.J.Weber,J.C.Morris,Adsorption Processes for Water Treatment,But-terworth,London,1987.[20]Q.Feng,Q.Lin,F.Gong,S.Sugita,M.Shoya,Adsorption of lead andmercury by rice husk ash,J.Colloid Interface Sci.278(2004)1–8. [21]M.Ahmaruzzaman,D.K.Sharma,Adsorption of phenols from wastewater,J.Colloid Interface Sci.287(2005)14–24.[22]O.Keskinkan,M.Z.L.Goksu,M.Basibuyuk,C.F.Forster,Heavy metaladsorption properties of a submerged aquatic plant(Ceratophyllum demer-sum),Bioresour.Technol.92(2004)197–200.[23]zaridis, D.D.Asouhidou,Kinetics of sorptive removal ofchromium(VI)from aqueous solutions by calcined Mg-Al-CO3hydrotal-cite,Water Res.37(2003)2875–2882.[24]D.H.Bangham,F.P.Burt,The behaviour of gases in contact with glasssurfaces,Proc.R.Soc.Lond.Ser.A–Containing Pap.A Mathemat.Phys.Character105(1924)481–488.[25]I.D.Mall,V.C.Srivastava,N.K.Agarwal,Removal of orange-G and methylviolet dyes by adsorption onto bagassefly ash kinetic study and equilibrium isotherm analyses,Dyes Pigments69(2006)210–223.[26]W.A.House,F.H.Denise,P.D.Armitage,Comparison of the uptake ofinorganic phosphorus to suspended and stream bed sediment,Water Res.29(1995)767–779.[27]R.-S.Juang,J.-Y.Shiau,Adsorption isotherms of phenols from water ontomacroreticular resins,J.Hazard.Mater.B70(1999)171–183.[28]Y.S.Ho,G.Mckay,Pseudo-second order model for sorption processes,Process Biochem.34(1999)451–465.[29]A.K.Battacharya,C.Venkobachar,Removal of cadmium(II)by low costadsorbents,J.Environ.Eng.Div.ASCE Proc.(1984)110–122.[30]K.V.Kumar,S.S.Ramamurthi,Adsorption of malachite green ontPithophora sp.a fresh water algae:Equilibrium and kinetic modelling, Process Biochem.40(2005)2865–2872.[31]S.Rengaraj,S.H.Moon,Kinetics of adsorption of Co(II)removal fromwater and wastewater by ion exchange resins,Water Res.36(2002) 1783–1793.[32]L.D.Michelson,P.G.Gideon,E.G.Pace,L.H.Kutal,Removal of Sol-uble Mercury from Wastewater by Complexing Techniques,Bull No.74,US.Dept.Industry,Office of Water Research and Technology, 1975.[33]J.Berrueta,J.M.Freije,G.Adrio,J.Coca,Synergistic effect in the mixturesof extraction of phenol from aqueous-solutions with n-butylacetate and acetophenone.Solvent extract,Ion Exchange8(1990)817–825.[34]M.K.Purkait,S.DasGupta,S.De,Adsorption of eosin dye on activatedcarbon and its surfactant based desorption,J.Environ.Manage.76(2005) 135–142.[35]M.Ajmal,R.A.Khan Rao,M.Ali Khan,Adsorption of copper from aque-ous solution on Brassica cumpestris(mustard oil cake),J.Hazard.Mater.B122(2005)177–183.[36]R.Naseem,S.S.Tahir,Removal of Pb(II)from aqueous/acidic solutionsby using bentonite as an adsorbent,Water Res.35(2001)3982–3986.。

Journal of Analytical and Applied Pyrolysis45(1998)153–169Kinetic study of scrap tyre pyrolysis and combustionD.Y.C.Leung *,C.L.WangUni 6ersity of Hong Kong ,Department of Mechanical Engineering ,7/F Haking Wong Building ,Pokfulam Road ,Hong Kong ,Hong KongReceived 12March 1997;accepted 16February 1998AbstractThis paper investigates the kinetic of pyrolysis and combustion of scrap tyre using thermogravimetric and derivative thermogravimetric analysis method.Three materials,namely tyre rubber powder,tyre fiber and wood powder were studied and compared with each other.The process parameters show that these three materials exhibit different thermal degradation patterns during pyrolysis and combustion process.Thermal degradation models were proposed to derive the kinetic parameters.It was found that the process and kinetic parameters vary with heating rates but are less dependent on the powder sizes.The simulations by the proposed models agreed well with experimental data.©1998Elsevier Science B.V.All rights reserved.Keywords :Pyrolysis;Combustion;Tyre powder;Tyre fiber;Wood powder;Char;Kinetic parameters;TGA;DTG;TG1.IntroductionThe disposal of used automotive tyres has caused many environmental and economical problems to most countries.In the US,750million to 2billion used tyres have been stockpiled,which are increasing at a rate of 280million per year [1].Also,15million tons of tyre are scraped every year in the European Union,2.5in North America,2.4in UK and 0.5in Japan [2].Most of the scrap tyres are dumped in open or landfill sites.As known,tyre is made of rubbery materials in the form of C x H y with some fibrous materials.It has high volatile and fixed carbon contents with heating value greater than that of coal.This makes it a good material for*Corresponding author.Tel.:+852********;fax:+852********;e-mail:ycleung@hkucc.hku.hk 0165-2370/98/$19.00©1998Elsevier Science B.V.All rights reserved.PII S0165-2370(98)00065-5154D.Y.C.Leung,C.L.Wang/J.Anal.Appl.Pyrolysis45(1998)153–169pyrolysis and combustion[3].On the other hand,scrap tyre is bulky and does not degrade in landfills.Therefore,open dumping of scrap tyre not only occupies a large space,presents an eyesore,causes potential health and environmental haz-ards,but also illustrates wastage of valuable energy resource.How to recover the energy efficiently from tyre is an acute and imperative problem to energy re-searchers.Pyrolysis,incineration,and gasification processes are considered to be more attractive and practicable methods for recovering energy from scrap tyre and biomass.Pyrolysis of carbonaceous materials can be interpreted as incomplete thermal degradation,generally in the absence of air,resulting in char,condensable liquids or tars,and trace amount of gaseous products.Gasification refers to pyrolysis followed by higher temperature reactions of the char,tars,and primary gases to yield mainly low molecular weight gaseous products.Several studies have been conducted to investigate the pyrolysis of scrap tyre and biomass in both laboratory and industrial scale.Boukadir et al.[4]found that the mechanism of rubber degradation is a two-step reaction with reaction order1 1.5for thefirst step and3for the second under isothermal conditions. However,they did notfind a correct activation energy.Bouvier et al.[5]reported that rubber degradation is a one-step mechanism and proposed to be afirst order reaction.The apparent activation energy and frequency factor were found to be 125.5kJ mol−1and1.08×109min−1using TGA.Kim et al.[6]proposed that each constituent of tyre might undergo an irreversiblefirst-order degradation independently.Chen and Yeh[7]investigated the Styrene-Butadiene Rubber using nitrogen as purge gas with different oxygen contents.They obtained apparent activation energy of211and153kJ mol−1,frequency factor of1.32×1014and 5.75×108min−1,and reaction order of0.6and0.48when the purge gas contains 0and20%oxygen,respectively.Xu and Wu[8]put forward a three-stage mechanism for the isothermal pyrolysis of wood powder and found that the reaction order,the apparent activation energy,and the frequency factor are 0.6–0.78,8.5–39.8kcal mol−1and40.8–2.66×109min−1,respectively at a temperature range of710–900°C.Urvan and Antal[9]studied the pyrolysis of sewage sludge,which produced volatile gas and solid residue.They modelled their data with two competitive reactions at heating rates from0.03–1.92°C min−1. High reaction orders(10and15,respectively)were found.Furthermore,it is noted that their model was not able tofit the data for different heating rates adequately.Ralf et al.[10]also studied the pyrolysis of sewage sludge by TGA and found lower reaction order(2and4,respectively)using a four-stage estima-tion.Other researchers have paid much attention to the weight loss characteristics during the pyrolysis of tyre or biomass[11–13]while the effects of powder size have not been studied in details.This paper aims to study the kinetic of the pyrolysis of different tyre powders and compare with that of wood powder,a good biomass material.The kinetic of the combustion of tyre rubber char and tyrefiber char was also studied.D.Y.C.Leung,C.L.Wang/J.Anal.Appl.Pyrolysis45(1998)153–169155 2.Experimental detailsThe physical,chemical and thermal characteristics of the samples are essential for the understanding of the sample behaviour in the pyrolysis process.The tyre powder and tyrefiber used in this study was supplied by the Recycling Plant of Guang Zhou in China while the wood powder by San Ya Timber Mill in China. The tyre powder sample was produced from granulating scrap tyre into four different sizes:1.18–2.36mm(8–16mesh),1.0–1.18mm(16mesh),0.5–0.6mm (30mesh)and0.355–0.425mm(40mesh).The typefiber is extracted from the scrap tyre during the production of the tyre powder.The size of wood powder used is0.075–0.088mm(180mesh).The proximate analysis of the tyre powder, tyrefiber and wood powder is shown in Table1which indicates that their compositions are quite different from one to another.The samples of tyre rubber char and tyrefiber char were prepared by heating the tyre rubber and tyrefiber in a stream of nitrogen gas with a temperature range of20–900°C at a heating rate of5°C min−1and soaking for1h at900°C in a Stanton Redcroft STA1500 Simultaneous Thermal Analyzer.The pyrolysis experiments of various samples and the combustion of tyre rubber char and tyrefiber char were conducted using the above mentioned thermal analyzer.About8mg of sample was placed in a platinum pan and heated in an inert atmosphere of nitrogen gas over a temperature range of 20–600°C at controlled heating rates of10,30,45and60°C min−1for pyrolysis. In the case of combustion test,it was conducted in a dry air atmosphere over a temperature range20–900°C at a heating rate of10°C min−1.The choice of the initial sample weight is based on the optimum kinetic rate controlled conditions. The sample weight loss percent(SWLP),sample temperature and heating effect were continuously recorded as a function of heating time.The heating effect was taken into account when recording sample temperature.So the SWLP can be a function of both sample temperature and heating time.From the SWLP,the normalized weight loss ratio(h)of sample can be obtained.The normalized weight loss rate(NWLR)of sample can be obtained by differentiating(8)with respect to time,which is a function of both sample temperature and heating time.Table1Composition of scrap tyre powder,tyrefiber and wood powderTyre rubberProximate analysis(%)Tyrefiber Wood powder Volatile80.064.268.1Fixed C 5.222.127.8Ash7.012.30.8Moisture1.02.59.0156D .Y .C .Leung ,C .L .Wang /J .Anal .Appl .Pyrolysis 45(1998)153–1693.Kinetic interpretation of experimental dataThe pyrolysis and combustion reactions could be represented by the following solid degradation equation derived by Vachuska and Voboril [14]:d h T d t =%n i =1d h id t =%ni =1K i (1−h i )(1)Since the degradation can be activated at a temperature lower than 800°C,which is within the temperature range of the kinetic reaction,the rate constant K i can be obtained from Arrehenius’Law:K i =A i exp (−E i /RT )(2)Using Eqs.(1)and (2),the kinetic parameters could be obtained from the thermogravimetry (TG)and derivative thermogravimetry (DTG)curves obtained through experiment.At the same time,the effect of reaction heat was introduced,which deviates from the sample heating rate by a constant.Thus the sample temperature changes with the actual heating rate.Substituting Eq.(2)into Eq.(1)and taking natural logarithm yieldsln d h T d t =ln %n i =1d h i d t =ln %ni =1A i exp −E i RT(1−h i )n(3)If there are two reactions occurring at different temperature regions,whichcorresponds to two compositional components degradation,the kinetic descriptions can be shown as:lnd h T d t =ln d h 1d t +d h 2d t=ln A 1exp −E 1RT (1−h 1)+A 2exp −E2RT(1−h 2)n(4)If only one reaction happens in a specific region,say reaction 1,then d h 2/d t =0,h 2=0and h 1=h T .Eq.(4)can thus be simplified tolnd h 1d t /(1−h 1)n=ln A 1−E1RT(5)Eq.(5)gives a straight line of slope E /R and a Y intercept of ln A when the lefthand side of the equation is plotted against 1/T .The value of h ,d h /d t and T at any times could be obtained from the experimental TG and DTG curves.Therefore,the kinetic parameters,i.e.,the apparent activation energy E and the frequency factor A ,could be determined from the above method.After that,the value of h 2and d h 2/d t could be obtained by subtracting h 1from h T and d h 1/d t from d h T /d t .Using the same method given above,the kinetic parameters of reaction 2can be obtained.157D .Y .C .Leung ,C .L .Wang /J .Anal .Appl .Pyrolysis 45(1998)153–1694.Theoretical prediction of and d /d tIf the kinetic parameters are known in addition to the temperature range and heating rate i ,the normalized weight loss ratio h could be calculated theoretically by integrating Eq.(1)for a temperature range of 0to T (K)as follows:&0hd h (1−h )=Ai &0T exp−ERTd T (6)The right-hand side of the above equation has no exact integral,but by using the relation of Coats and Redfern [15],it can be approximated as:A i&0T exp−E RT d T =A i RT 2E exp −ERT%k =0(−2)k(E /RT )k Hence,h can be expressed as:h =1−exp −A RT 2E exp −ERT%k =0(−2)k(E /RT )k n(7)From the above equations,the pyrolysis process parameters,such as h and d h /d t ,could be predicted theoretically.5.Results and discussion5.1.Tyre rubber pyrolysis and tyre rubber char combustionFig.1a and Fig.2show the typical DTG curves for the pyrolysis of tyre rubber of size 40mesh at various heating rates and of various mesh sizes at a heating rate of 10°C min −1,respectively.Table 2shows the process parameters of tyre rubber pyrolysis according to different degradation temperature regions.It is interesting to note that the DTG curve exhibits three different NWLR regions over a temperature range of 150–600°C.These characteristics may be due to the fact that the main constituents of tyre used in the present study are either nature rubber (NR),styrene butadiene rubber (SBR),butadiene rubber (BR)or their combination with mois-ture,oil,plasticizer and additives as minor constituents.All these constituents lose their weight at different rates and at different temperatures.According to the evaporating characteristics of individual constituents,it can be deduced that the moisture inside the type powder evaporates before the temperature reached 150°C.At the temperature range of 150–350°C,the oil,plasticizer and additives are lost.The loss of NR,SBR and BR or their combination at the temperature range of 340–550°C gives two peaks in the NWLR curve at about 380and 450°C.Liu et al.[16]investigated the pyrolysis of NR,BR and SBR and found that the maximum weight loss rate of NR occurs at a temperature of 373°C,BR at 372°C and 460°C,SBR at 372°C and 429–460°C.Their results indicated that the three rubber materials contribute their weight loss together over the temperature range of 370to 460°C,which matches with the present result.158D .Y .C .Leung ,C .L .Wang /J .Anal .Appl .Pyrolysis 45(1998)153–169T a b l e 2A n a l y z e d r e s u l t s o f t y r e p o w d e r ,t y r e fib e r a n d w o o d p o w d e r p y r o l y s i sP y r o l y s i s p r o c e s s p a r a m e t e r sS a m p l e n a m eH e a t i n g r a t e T e m p .a t T o t a l w e i g h t T e m p .a t F i n i s h i n g W e i g h t l o s s R e a c t i o n N W L R a t W e i g h t l o s s N W L R a t S t a r t i n g S a m p l e s i z e t i m e a t p e a k o n e (°C m i n −1)p e a k o n e l o s s p e a k t w o p e a k t w o a t p e a k t w o (m e s h n o .)t e m p .t e m p .p e a k o n e (°C )(°C )(°C )(%)(m i n −1)(m i n −1)(%)(m i n )(°C )(%)50029.90.10760.062128.1538356.9118568.71450T y r e p o w d e r 108 1650029.90.09320.074222.8455.3965.841618538045850029.90.10200.066823.3656.6945865.263791853045050029.90.10780.060625.7853.3463.74401853815208.50.30450.194920.5140053.9248064.402508 16305208.50.31410.185324.1656.3966.73162504044795208.50.31070.181921.8653.9247863.71401250304715208.50.32990.170022.9250.1761.41402504045455.50.5873—25.21410—66.45—2608 16454915455.50.48820.296820.1254.6365.22162604115455.50.49420.273619.4252.2363.62302604094855255.50.52160.248723.1051.5848461.73414260405454.20.81850.412517.2052.416064.158 162604104925454.20.73000.374419.4951.5748563.13410260165454.20.77290.348522.2553.3263.42302604164905454.20.78040.331320.6850.1648460.524132604051020.30.0167—42.10T y r e fib e r—1082.59295412—5256.70.0169—42.49——83.95445310—30447—5354.80.0152—34.76—80.00453005353.50.0219—34.58——81.88604503003471069a 40021.50.00890.0018a50.284.81a76.51180W o o d p o w -d e r4206.80.00820.0017a50.774.29a74.4030—20036884a4204.30.00800.0019a50.014.34a88a74.20369452004403.2600.00840.0015a49.553.94a72.2922038192aaR e p r e s e n t i n g t h e d a t e o f w a t e r m o i s t u r e s e p a r a t e d o u t f r o m t h e w o o d p o w d e rD.Y.C.Leung,C.L.Wang/J.Anal.Appl.Pyrolysis45(1998)153–169159Fig.1.Normalized weight loss rate against temperature for pyrolysis of different materials at different heating rates.(a)Tyre powder;(b)tyrefiber;and(c)wood powder.160D.Y.C.Leung,C.L.Wang/J.Anal.Appl.Pyrolysis45(1998)153–169Fig.1.(Continued)Yang et al.[17]also studied the pyrolysis behaviour of these three rubbers and their mixture.The curves they reported at a heating rate of10°C min−1showed that two peaks of the NWLR are observed for each rubber.Thefirst peak,which has a maximum weight loss rate at245°C for NR and BR,and267°C for SBR,is attributed to the volatilization of the processing oil,plasticizer,moisture,and any other low boiling point components.The second peak,which has a maximum weight loss rate at377°C for NR,465°C for BR,and444°C for SBR,is attributed to the three straight elastomers thermal decomposition.In addition,they obtained several DTG curves for the pyrolysis of the mixture with different compositions, which are similar to those obtained presently.In addition,Brazier and Nickel[18]also reported a similar DTG curve of tyre rubber pyrolysis,consisting of only two main components of NR and BR.Kim et al.[6]investigated the pyrolysis process of sidewall and tread rubber derived from waste tyre at different heating rates using TGA.Their DTG curves are different from the present curves,which may be due to the difference in the sample composition.Nevertheless,they also divided the degradation process into three regions with different temperature ranges in explaining the pyrolysis process. It is well understood that many complex reactions are involved in the pyrolysis process of tyre rubber.Therefore,it is impossible to develop a precise kinetic model for determining various kinetic parameters from thermogravimetric data alone.It is generally accepted that the important parameters in pyrolysis are temperature, sample weight and its loss rate,time,and heating rates.Based on these parameters161D.Y.C.Leung,C.L.Wang/J.Anal.Appl.Pyrolysis45(1998)153–169Fig.2.Normalized weight loss rate against temperature for pyrolysis of different sizes of tyre rubber at a heating rate of10°C min−1.a model is proposed to predict the weight and weight loss rate profiles of tyre rubber.The model assumes two reactions in the tyre rubber pyrolysis,correspond-ing to the main components(NR,BR and SBR)degradation.Each component contributes to the decomposition at different temperature regions forming volatiles and char.The pyrolysis rate is considered to be the sum of the two reaction rates.A schematic diagram of the model is shown below:It is further assumed that reactions1and2follow the Arrehenius’Law for ascertaining the values of the rate constant and undergo an irreversiblefirst order degradation independently.Furthermore,reaction1mainly occurs at a lower temperature region than reaction2at a higher temperature region and both reactions occur at the intermediate temperature region.Fig.4shows the typical DTG curves of tyre rubber char of different mesh sizes at a heating rate of10°C min−1.It can be found that the char degrades in one temperature region ing the same approach as mentioned above to investi-gate the behaviour of tyre rubber char combustion,and assuming that only one162D .Y .C .Leung ,C .L .Wang /J .Anal .Appl .Pyrolysis 45(1998)153–169T a b l e 3K i n e t i c p a r a m e t e r s o f t h e p y r o l y s i s o f t y r e p o w d e r ,t y r e fib e r a n d w o o d p o w d e rH i g h e r t e m p e r a t u r e s t a g eH e a t i n g r a t e S a m p l eL o w e r t e m p e r a t u r e s t a g e (°C m i n −1)F r e q u e n c y f a c t o r T e m p e r a t u r e A p p a r e n t a c t i v a -A p p a r e n t a c t i v a -T e m p e r a t u r e F r e q u e n c y f a c t o r (m i n −1)t i o n e n e r g y (m i n −1)r a n g e (°C )r a n g e (°C )t i o n e n e r g y (k J m o l −1)(k J m o l −1)350 500136.1102.31×109300 420T y r e p o w d e r 164.56.29×1013(40m e s h )133.62.09×109370 510310 440180.9301.32×1014400 54045107.03.34×107320 470203.47.58×10151.13×1017410 54099.11.02×10760320 480218.71.39×106240 400112.71.23×10910W o o d p o w d e r150 34070.0119.16.50×109270 410307.46×10882.8150 350270 41045134.81.53×1011150 350102.74.47×109280 42060141.45.56×1011160 360104.95.75×1094.81×1011152.0320 50010T y r e fib e r6.91×101130340 520160.7163.845340 5206.91×101160350 520201.15.11×1014reaction occurs during char combustion,we propose the following model to describe the combustion process:Tyre Rubber Char K1Ash+GasesIn this model,the reaction is treated as one component undergoing an irre-versiblefirst order degradation,which follows the Arrehenius’Law for ascertaining the values of the rate constant.According to the above model,the kinetic parameters of tyre rubber pyrolysis and tyre rubber char combustion are obtained and shown in Tables3and4.Yang et al.[15]reported the kinetic parameters of NR,BR and SBR pyrolysis at a heating rate of10°C min−1.The activation energy is207,215and152kJ mol−1 and the frequency factor is2.36×1016,6.32×1014and4.15×1010min−1for NR, BR and SBR,respectively,which are slightly higher than those obtained from our model(Table3).This may be due to the fact that the kinetic parameters obtained from our model are the combined effect of these three rubbers.In addition,Kim et al.[6]reported the kinetic parameters of sidewall and tread rubber pyrolysis according to three temperature regions.For sidewall pyrolysis,the average apparent activation energy and frequency factor were204kJ mol−1and 2.04×1014min−1for components decomposed at higher temperature,195kJ mol−1and2.08×1015min−1for components decomposed at medium temperature, 42kJ mol−1and1436min−1for components decomposed at lower temperature. The average apparent activation energy and frequency factor were127,209,39kJ mol−1and8.75×108,3.78×1016and934.5min−1for components of tread rubber decomposed at higher,medium and lower temperature.These values are only slightly different from the present result.The simulation curves are shown in Fig.3a and Fig.4.Scrutinizing the results obtained,it can be observed that the total weight loss during the pyrolysis process is 60–69%,among them only 7%is due to moisture,oil,plasticizer and additives, 44–53%and 40–49%is lost in reactions1and2,respectively. During the tyre rubber char combustion process,the total weight loss is 84–93%. Furthermore,the following points were noted after analyzing the results:The sizes of tyre powder have little effect on the pyrolysis and combustion process;The heating rate affects the pyrolysis significantly;Table4Kinetic parameters of the combustion of tyre rubber char and tyrefiber charSample size Apparent activation energySample nameFrequency factorTemperature(mesh)(min−1)(kJ mol−1)range(°C)8–16450 610Tyre powder 2.89×108145.416148.1 3.84×108450 62040450 620161.2 3.93×109Tyrefiber237.39.43×1014480 570Fig.3.Theoretical and experimental normalized weight loss rate against temperature for the pyrolysis of different materials at a heating rate of 10°C min −1.(a)Tyre powder;(b)tyre fiber;and (c)wood powder.Fig.3.(Continued)As the heating rate rises,(a)the reaction shifts to higher temperature range,i.e.the starting and ending temperature increase;(b)the temperature corresponding to the peak value of NWLR increases;(c)the NWLR increases and the reaction time decreases dramatically.During pyrolysis the apparent activation energy and frequency factor of reaction 1increase with heating rate,indicating that the degradation is more difficult.In contrast,the apparent activation energy and frequency factor of pyrolysis reaction2decrease,making it easier to degrade.The apparent activation energy and frequency factor of combustion of tyre rubber char increases as the size of char increases.However,the difference is not significant.The pyrolysis of tyre rubber and the combustion of its char occurs at tempera-ture ranges of about150–500°C and460–620°C,respectively.There is some overlap in the temperature range,so it can be assumed that these two reactions follow smoothly.5.2.Tyrefiber pyrolysis and tyrefiber char combustionFig.1b and Fig.3b show the typical DTG curves for the tyrefiber pyrolysis at various heating rates and tyrefiber char combustion at a heating rate of10°C min−1,respectively.Table2shows the detailed process parameters of tyrefiber pyrolysis.It can be found that the degradation occurs at the temperature region of350–520°C for pyrolysis and500–570°C for combustion.This behaviour is similarparison of theoretical and experimental normalized weight loss rate against temperature for the combustion of tyre rubber char.to that of tyre rubber char combustion,so the same model as mentioned in tyre rubber char combustion is ing this model,the kinetic parameters are obtained and shown in Tables3and4.The simulation curves are shown in Fig.3b. It can be found that the model simulationfits the experimental data well.The following points were observed from the experimental results.The heating rate affects the pyrolysis significantly;During pyrolysis,as the heating rate rises,(a)the reaction shifts to higher temperature range;(b)the temperature corresponding to peak value of NWLR increases;(c)the NWLR increases and the reaction time decreases dramatically;(d)the apparent activation energy and frequency factor increase,which leads toa more difficult degradation.The total weight loss is 80–84%for tyrefiber pyrolysis and 71%for tyre fiber char combustion.The pyrolysis of tyrefiber and the combustion of its char occurs at different temperature ranges which overlap slightly so it can be assumed that these two reactions are carried out continuously.For the same time period,about82%of total weight loss occurred in pyrolysis and only about12%in combustion.Furthermore,the apparent activation energy and frequency factor is larger in combustion than in pyrolysis process.This indicates that attention must be paidto tyrefiber char combustion.5.3.Wood powder pyrolysisFig.1c shows typical DTG curves for wood powder pyrolysis at various heating rates.Table2shows the detailed process parameters according to differ-ent degradation temperature paring the TG and DTG curves with that of tyre rubber in Fig.1a,it was found that the curve also shows three different NWLR regions over a temperature range of20–420°C.It is well known that wood is consisted of three main constituents:moisture,lignin and holocellu-lose(cellulose and hemicellulose).The characteristics of wood degradation at different temperature regions are made up of these three components.For the wood powder pyrolysis at a heating rate of10°C min−1,the moisture inside the wood powder evaporates at the temperature range of20–150°C.At the tempera-ture range of200–420°C,the loses of lignin and holocellulose give two peaks in the NWLR curve at about300°C and340°C.Also,it can be found that the temperature corresponding to the peaks value of NWLR are affected by heating rates.There are several papers investigating the pyrolysis of cellulose,biomass and wood powder[8,19–21]and different models were proposed to simulate the pyrolysis process.From the present results,we used a similar model as that used in tyre rubber pyrolysis to explain the wood powder pyrolysis which is given below:From the above model,the kinetic parameters and simulation curves are ob-tained and shown in Table3and Fig.3c,respectively.It can be seen that for the total weight loss,only 9%is due to moisture,and17–22%,69–74%is due to reaction1and reaction2,respectively.In addition,the apparent activation energy and frequency factor for moisture evaporation are31,47kJ mol−1and2.02×104,5.99×106min−1at the heating rate of10and60°C min−1,respectively.In general,as the heating rate increases,the apparent activation energy and fre-quency factor increase significantly.It could be found that the kinetic parameters obtained match well with those of Xu and Wu[8]despite the fact that the tests were carried out under different conditions.It should be emphasized that although it is difficult to compare our results with those in literature directly due to the different tyre samples and experimental conditions,the good matching of the model simulation with experimental data indicates that the models can be used to describe the pyrolysis and combustion process of the tyre rubber and tyrefibre.6.ConclusionsThe thermogravimetric and derivative thermogravimetric analysis conducted in this study provided valuable information on the pyrolysis kinetics and mechanisms of heterogeneous materials like tyre rubber powder,tyrefiber and wood powder. The pyrolysis of tyre rubber exhibits three obvious weight loss regions occurring at temperature ranges of about150–350°C,330–450°C and420–520°C,respectively while the combustion of tyre rubber char has one obvious weight loss region occurring at a temperature range of450–620°C.The tyrefiber pyrolysis and its char combustion have one obvious weight loss region,which occurs at a temperature range of about350–520°C and500–570°C,respectively.Similar to the case of tyre rubber,the pyrolysis of wood powder has three obvious weight loss regions occurring at temperature ranges of about20–150°C,150–360°C and240–420°C, respectively.Due to these observations,pyrolysis and combustion models were proposed to study the kinetic of the processes,which were found tofit the experimental data well.It was also found that the heating rate has a significant effect on the pyrolysis and combustion process.With increasing heating rate the weight loss regions shift to a higher temperature range and the weight loss rate is increased.The reaction time shortened quickly but the total weight loss has no obvious change.The apparent activation energy and frequency factor also increase with increasing in heating rate, hence increasing the difficulties of the pyrolysis reaction.From the activation energy obtained,the pyrolysis of tyre rubber was found to be easier than tyrefiber but more difficult than wood powder.Moreover,the combustion of tyre rubber char is found to be easier than tyrefiber char.The size of tyre rubber and its char was found to produce no significant effect on the pyrolysis and combustion process. AcknowledgementsThe authors wish to acknowledge the Honk Kong Research Grant Council and the CRCG of the University of Hong Kong for supporting this project. Appendix A.NomenclatureAfrequency factor(min−1)Eapparent activation energy(kJ mol−1)rate constant(min−1)KR gas constant,R=8.314×10−3(kJ mol−1·K−1)treaction time(min)Tsample temperature(K)W0sample weight at start time(mg)Wsample weight at time t(mg)W sample weight at end time(mg)。

网络出版时间：2013-08-23 13:08网络出版地址：/kcms/detail/11.2206.TS.20130823.1308.007.html2013-08-20动力学模型预测药膳型老鸭煲货架期谢程炜1,2,诸永志1*,王道营1,刘芳1,徐为民1( 1. 江苏省农业科学院农产品加工研究所，江苏南京210014; 2. 扬州大学食品科学与工程学院，江苏扬州225127)摘要：为了确定药膳型老鸭煲在常温（25℃）下的货架期，本文先对杀菌温度进行选择，并对最优杀菌温度杀菌后的药膳型老鸭煲采用加速货架期法，结合感官评定，以硫代巴比妥酸（TBA）值为指标建立了脂肪氧化的一级反应动力学方程。

结果表明，121℃下杀菌效果最好，感官评定与TBA值之间具有极显著的相关性，动力学方程拟合度高（相关系数R2在0.99以上）。

由动力学方程计算出鸭肉包在常温下的货架期为301天，汤包的货架期为282天，油包的货架期为213天，由此推测出药膳型老鸭煲在常温下的货架期为213天。

关键词：老鸭煲；货架期；感官评定；TBA值；动力学方程Modeling the Shelf life of Medicated Old Duck PotXIE Cheng-wei1,2,ZHU Yong-zhi1*,WANG Dao-ying1,LIU Fang1,XU Wei-min1(1. Institute of Agricultural Products Processing, Jiangsu Academy of Agricultural Sciences, Nanjing 210014, China; 2. Collegeof Food Science and Engineering, Yangzhou University, Yangzhou 225127, China)Abstract: In order to obtain the shelf life of medicated old duck pot at room temperature (25℃), the sterilization temperature was selected first, then accelerated shelf life test was tested after sterilizing with the optimal sterilization temperatureby combination of sensory evaluation, and one order chemical reaction kinetic equation according to TBA values was established.. The results showed that the best sterilization temperature was 121°C, and there was a highly significant correlationbetween sensory evaluation and TBA values. The correlation coefficient of reaction kinetic equation was greater than 0.99. Theshelf life of the package of duck, soup and oil was 301 days, 282 days and 213 days respectively at room temperature, whichcalculated by the kinetic equation. Conclude that the shelf life of medicated old duck pot was 213 days at room temperature.Key Words: Old duck pot; Shelf life; Sensory evaluation; TBA values; Kinetic equation中图分类号：TS251.6+8 文献标志码：A 文章编号：食品货架期受食品中的微生物、酶类和生化反应的影响[1]。

参数灵敏度分析流程英文回答：Sensitivity analysis is a technique that is used to determine how changes in the value of a parameter or set of parameters in a mathematical model can affect the output or outcome of the model. It is an important tool in various fields such as engineering, finance, and environmental science, where decision-making relies on the accuracy and reliability of mathematical models.The process of conducting sensitivity analysis involves the following steps:1. Identify the parameters: The first step is to identify the parameters in the model that are of interest and have the potential to impact the model's output. These parameters can be inputs, constants, or variables.2. Define the range of variation: Once the parametersare identified, the next step is to define the range of variation for each parameter. This can be done by specifying minimum and maximum values or by using probability distributions if the parameter is uncertain.3. Select the sensitivity measure: There are different sensitivity measures that can be used to quantify the impact of parameter variations on the model output. Some commonly used measures include the absolute sensitivity index, relative sensitivity index, and the variance-based sensitivity index.4. Perform the analysis: With the parameters, variation range, and sensitivity measure defined, the sensitivity analysis can be performed. This typically involves running the model multiple times with different parameter values within the specified range and recording the corresponding outputs.5. Analyze the results: Once the sensitivity analysisis complete, the results need to be analyzed to determine the sensitivity of the model output to each parameter. Thiscan be done by examining the sensitivity measures and identifying the most influential parameters.6. Interpret the findings: The final step is to interpret the findings of the sensitivity analysis and draw conclusions. This may involve identifying key parameters that significantly impact the model output, assessing the robustness of the model, and identifying areas wherefurther research or data collection is needed.参数灵敏度分析是一种用于确定数学模型中参数或一组参数的值变化如何影响模型输出或结果的技术。

物理化学概念及术语A B C D E F G H I J K L M N O P Q R S T U V W X Y Z概念及术语 (16)BET公式BET formula (16)DLVO理论 DLVO theory (16)HLB法hydrophile-lipophile balance method (16)pVT性质 pVT property (16)ζ电势 zeta potential (16)阿伏加德罗常数 Avogadro’number (16)阿伏加德罗定律 Avogadro law (16)阿累尼乌斯电离理论Arrhenius ionization theory (16)阿累尼乌斯方程Arrhenius equation (17)阿累尼乌斯活化能 Arrhenius activation energy (17)阿马格定律 Amagat law (17)艾林方程 Erying equation (17)爱因斯坦光化当量定律 Einstein’s law of photochemical equivalence (17)爱因斯坦－斯托克斯方程 Einstein-Stokes equation (17)安托万常数 Antoine constant (17)安托万方程 Antoine equation (17)盎萨格电导理论Onsager’s theory of conductance (17)半电池half cell (17)半衰期half time period (18)饱和液体 saturated liquids (18)饱和蒸气 saturated vapor (18)饱和吸附量 saturated extent of adsorption (18)饱和蒸气压 saturated vapor pressure (18)爆炸界限 explosion limits (18)比表面功 specific surface work (18)比表面吉布斯函数 specific surface Gibbs function (18)比浓粘度 reduced viscosity (18)标准电动势 standard electromotive force (18)标准电极电势 standard electrode potential (18)标准摩尔反应焓 standard molar reaction enthalpy (18)标准摩尔反应吉布斯函数 standard Gibbs function of molar reaction (18)标准摩尔反应熵 standard molar reaction entropy (19)标准摩尔焓函数 standard molar enthalpy function (19)标准摩尔吉布斯自由能函数 standard molar Gibbs free energy function (19)标准摩尔燃烧焓 standard molar combustion enthalpy (19)标准摩尔熵 standard molar entropy (19)标准摩尔生成焓 standard molar formation enthalpy (19)标准摩尔生成吉布斯函数 standard molar formation Gibbs function (19)标准平衡常数 standard equilibrium constant (19)标准氢电极 standard hydrogen electrode (19)标准态 standard state (19)标准熵 standard entropy (20)标准压力 standard pressure (20)标准状况 standard condition (20)表观活化能apparent activation energy (20)表观摩尔质量 apparent molecular weight (20)表观迁移数apparent transference number (20)表面 surfaces (20)表面过程控制 surface process control (20)表面活性剂surfactants (21)表面吸附量 surface excess (21)表面张力 surface tension (21)表面质量作用定律 surface mass action law (21)波义尔定律 Boyle law (21)波义尔温度 Boyle temperature (21)波义尔点 Boyle point (21)玻尔兹曼常数 Boltzmann constant (22)玻尔兹曼分布 Boltzmann distribution (22)玻尔兹曼公式 Boltzmann formula (22)玻尔兹曼熵定理 Boltzmann entropy theorem (22)泊Poise (22)不可逆过程 irreversible process (22)不可逆过程热力学thermodynamics of irreversible processes (22)不可逆相变化 irreversible phase change (22)布朗运动 brownian movement (22)查理定律 Charle’s law (22)产率 yield (23)敞开系统 open system (23)超电势 over potential (23)沉降 sedimentation (23)沉降电势 sedimentation potential (23)沉降平衡 sedimentation equilibrium (23)触变 thixotropy (23)粗分散系统 thick disperse system (23)催化剂 catalyst (23)单分子层吸附理论 mono molecule layer adsorption (23)单分子反应 unimolecular reaction (23)单链反应 straight chain reactions (24)弹式量热计 bomb calorimeter (24)道尔顿定律 Dalton law (24)道尔顿分压定律 Dalton partial pressure law (24)德拜和法尔肯哈根效应Debye and Falkenhagen effect (24)德拜立方公式 Debye cubic formula (24)德拜－休克尔极限公式 Debye-Huckel’s limiting equation (24)等焓过程 isenthalpic process (24)等焓线isenthalpic line (24)等几率定理 theorem of equal probability (24)等温等容位Helmholtz free energy (25)等温等压位Gibbs free energy (25)等温方程 equation at constant temperature (25)低共熔点 eutectic point (25)低共熔混合物 eutectic mixture (25)低会溶点 lower consolute point (25)低熔冰盐合晶 cryohydric (26)第二类永动机 perpetual machine of the second kind (26)第三定律熵 Third-Law entropy (26)第一类永动机 perpetual machine of the first kind (26)缔合化学吸附 association chemical adsorption (26)电池常数 cell constant (26)电池电动势 electromotive force of cells (26)电池反应 cell reaction (27)电导 conductance (27)电导率 conductivity (27)电动势的温度系数 temperature coefficient of electromotive force (27)电动电势 zeta potential (27)电功electric work (27)电化学 electrochemistry (27)电化学极化 electrochemical polarization (27)电极电势 electrode potential (27)电极反应 reactions on the electrode (27)电极种类 type of electrodes (27)电解池 electrolytic cell (28)电量计 coulometer (28)电流效率current efficiency (28)电迁移 electro migration (28)电迁移率 electromobility (28)电渗 electroosmosis (28)电渗析 electrodialysis (28)电泳 electrophoresis (28)丁达尔效应 Dyndall effect (28)定容摩尔热容 molar heat capacity under constant volume (28)定容温度计 Constant voIume thermometer (28)定压摩尔热容 molar heat capacity under constant pressure (29)定压温度计 constant pressure thermometer (29)定域子系统 localized particle system (29)动力学方程kinetic equations (29)动力学控制 kinetics control (29)独立子系统 independent particle system (29)对比摩尔体积 reduced mole volume (29)对比体积 reduced volume (29)对比温度 reduced temperature (29)对比压力 reduced pressure (29)对称数 symmetry number (29)对行反应reversible reactions (29)对应状态原理 principle of corresponding state (29)多方过程polytropic process (30)多分子层吸附理论 adsorption theory of multi-molecular layers (30)二级反应second order reaction (30)二级相变second order phase change (30)法拉第常数 faraday constant (31)法拉第定律 Faraday’s law (31)反电动势back E.M.F. (31)反渗透 reverse osmosis (31)反应分子数 molecularity (31)反应级数 reaction orders (31)反应进度 extent of reaction (32)反应热heat of reaction (32)反应速率rate of reaction (32)反应速率常数 constant of reaction rate (32)范德华常数 van der Waals constant (32)范德华方程 van der Waals equation (32)范德华力 van der Waals force (32)范德华气体 van der Waals gases (32)范特霍夫方程 van’t Hoff equation (32)范特霍夫规则 van’t Hoff rule (33)范特霍夫渗透压公式 van’t Hoff equation of osmotic pressure (33)非基元反应 non-elementary reactions (33)非体积功 non-volume work (33)非依时计量学反应 time independent stoichiometric reactions (33)菲克扩散第一定律 Fick’s first law of diffusion (33)沸点 boiling point (33)沸点升高 elevation of boiling point (33)费米－狄拉克统计Fermi-Dirac statistics (33)分布 distribution (33)分布数 distribution numbers (34)分解电压 decomposition voltage (34)分配定律 distribution law (34)分散系统 disperse system (34)分散相 dispersion phase (34)分体积 partial volume (34)分体积定律 partial volume law (34)分压 partial pressure (34)分压定律 partial pressure law (34)分子反应力学 mechanics of molecular reactions (34)分子间力 intermolecular force (34)分子蒸馏molecular distillation (35)封闭系统 closed system (35)附加压力 excess pressure (35)弗罗因德利希吸附经验式 Freundlich empirical formula of adsorption (35)负极 negative pole (35)负吸附 negative adsorption (35)复合反应composite reaction (35)盖.吕萨克定律 Gay-Lussac law (35)盖斯定律 Hess law (35)甘汞电极 calomel electrode (35)感胶离子序 lyotropic series (35)杠杆规则 lever rule (35)高分子溶液 macromolecular solution (36)高会溶点 upper consolute point (36)隔离法the isolation method (36)格罗塞斯－德雷珀定律 Grotthus-Draoer’s law (36)隔离系统 isolated system (37)根均方速率 root-mean-square speed (37)功 work (37)功函work content (37)共轭溶液 conjugate solution (37)共沸温度 azeotropic temperature (37)构型熵configurational entropy (37)孤立系统 isolated system (37)固溶胶 solid sol (37)固态混合物 solid solution (38)固相线 solid phase line (38)光反应 photoreaction (38)光化学第二定律 the second law of actinochemistry (38)光化学第一定律 the first law of actinochemistry (38)光敏反应 photosensitized reactions (38)光谱熵 spectrum entropy (38)广度性质 extensive property (38)广延量 extensive quantity (38)广延性质 extensive property (38)规定熵 stipulated entropy (38)过饱和溶液 oversaturated solution (38)过饱和蒸气 oversaturated vapor (38)过程 process (39)过渡状态理论 transition state theory (39)过冷水 super-cooled water (39)过冷液体 overcooled liquid (39)过热液体 overheated liquid (39)亥姆霍兹函数 Helmholtz function (39)亥姆霍兹函数判据 Helmholtz function criterion (39)亥姆霍兹自由能 Helmholtz free energy (39)亥氏函数 Helmholtz function (39)焓 enthalpy (39)亨利常数 Henry constant (39)亨利定律 Henry law (39)恒沸混合物 constant boiling mixture (40)恒容摩尔热容 molar heat capacity at constant volume (40)恒容热 heat at constant volume (40)恒外压 constant external pressure (40)恒压摩尔热容 molar heat capacity at constant pressure (40)恒压热 heat at constant pressure (40)化学动力学chemical kinetics (40)化学反应计量式 stoichiometric equation of chemical reaction (40)化学反应计量系数 stoichiometric coefficient of chemical reaction (40)化学反应进度 extent of chemical reaction (41)化学亲合势 chemical affinity (41)化学热力学chemical thermodynamics (41)化学势 chemical potential (41)化学势判据 chemical potential criterion (41)化学吸附 chemisorptions (41)环境 environment (41)环境熵变 entropy change in environment (41)挥发度volatility (41)混合熵 entropy of mixing (42)混合物 mixture (42)活度 activity (42)活化控制 activation control (42)活化络合物理论 activated complex theory (42)活化能activation energy (43)霍根-华森图 Hougen-Watson Chart (43)基态能级 energy level at ground state (43)基希霍夫公式 Kirchhoff formula (43)基元反应elementary reactions (43)积分溶解热 integration heat of dissolution (43)吉布斯－杜亥姆方程 Gibbs-Duhem equation (43)吉布斯－亥姆霍兹方程 Gibbs-Helmhotz equation (43)吉布斯函数 Gibbs function (43)吉布斯函数判据 Gibbs function criterion (44)吉布斯吸附公式Gibbs adsorption formula (44)吉布斯自由能 Gibbs free energy (44)吉氏函数 Gibbs function (44)极化电极电势 polarization potential of electrode (44)极化曲线 polarization curves (44)极化作用 polarization (44)极限摩尔电导率 limiting molar conductivity (44)几率因子 steric factor (44)计量式 stoichiometric equation (44)计量系数 stoichiometric coefficient (45)价数规则 rule of valence (45)简并度 degeneracy (45)键焓bond enthalpy (45)胶冻 broth jelly (45)胶核 colloidal nucleus (45)胶凝作用 demulsification (45)胶束micelle (45)胶体 colloid (45)胶体分散系统 dispersion system of colloid (45)胶体化学 collochemistry (45)胶体粒子 colloidal particles (45)胶团 micelle (45)焦耳Joule (45)焦耳-汤姆生实验 Joule-Thomson experiment (46)焦耳-汤姆生系数 Joule-Thomson coefficient (46)焦耳-汤姆生效应 Joule-Thomson effect (46)焦耳定律 Joule's law (46)接触电势contact potential (46)接触角 contact angle (46)节流过程 throttling process (46)节流膨胀 throttling expansion (46)节流膨胀系数 coefficient of throttling expansion (46)结线 tie line (46)结晶热heat of crystallization (47)解离化学吸附 dissociation chemical adsorption (47)界面 interfaces (47)界面张力 surface tension (47)浸湿 immersion wetting (47)浸湿功 immersion wetting work (47)精馏 rectify (47)聚（合）电解质polyelectrolyte (47)聚沉 coagulation (47)聚沉值 coagulation value (47)绝对反应速率理论 absolute reaction rate theory (47)绝对熵 absolute entropy (47)绝对温标absolute temperature scale (48)绝热过程 adiabatic process (48)绝热量热计adiabatic calorimeter (48)绝热指数 adiabatic index (48)卡诺定理 Carnot theorem (48)卡诺循环 Carnot cycle (48)开尔文公式 Kelvin formula (48)柯诺瓦洛夫－吉布斯定律 Konovalov-Gibbs law (48)科尔劳施离子独立运动定律 Kohlrausch’s Law of Independent Migration of Ions (48)可能的电解质potential electrolyte (49)可逆电池 reversible cell (49)可逆过程 reversible process (49)可逆过程方程 reversible process equation (49)可逆体积功 reversible volume work (49)可逆相变 reversible phase change (49)克拉佩龙方程 Clapeyron equation (49)克劳修斯不等式 Clausius inequality (49)克劳修斯－克拉佩龙方程 Clausius-Clapeyron equation (49)控制步骤 control step (50)库仑计 coulometer (50)扩散控制 diffusion controlled (50)拉普拉斯方程 Laplace’s equation (50)拉乌尔定律 Raoult law (50)兰格缪尔－欣谢尔伍德机理 Langmuir-Hinshelwood mechanism (50)雷利公式 Rayleigh equation (50)兰格缪尔吸附等温式 Langmuir adsorption isotherm formula (50)冷冻系数coefficient of refrigeration (50)冷却曲线 cooling curve (51)离解热heat of dissociation (51)离解压力dissociation pressure (51)离域子系统 non-localized particle systems (51)离子的标准摩尔生成焓 standard molar formation of ion (51)离子的电迁移率 mobility of ions (51)离子的迁移数 transport number of ions (51)离子独立运动定律 law of the independent migration of ions (51)离子氛 ionic atmosphere (51)离子强度 ionic strength (51)理想混合物 perfect mixture (52)理想气体 ideal gas (52)理想气体的绝热指数 adiabatic index of ideal gases (52)理想气体的微观模型 micro-model of ideal gas (52)理想气体反应的等温方程 isothermal equation of ideal gaseous reactions (52)理想气体绝热可逆过程方程 adiabatic reversible process equation of ideal gases (52)理想气体状态方程 state equation of ideal gas (52)理想稀溶液 ideal dilute solution (52)理想液态混合物 perfect liquid mixture (52)粒子 particles (52)粒子的配分函数 partition function of particles (53)连串反应consecutive reactions (53)链的传递物 chain carrier (53)链反应 chain reactions (53)量热熵 calorimetric entropy (53)量子统计quantum statistics (53)量子效率 quantum yield (53)临界参数 critical parameter (53)临界常数 critical constant (53)临界点 critical point (53)临界胶束浓度critical micelle concentration (53)临界摩尔体积 critical molar volume (54)临界温度 critical temperature (54)临界压力 critical pressure (54)临界状态 critical state (54)零级反应zero order reaction (54)流动电势 streaming potential (54)流动功 flow work (54)笼罩效应 cage effect (54)路易斯－兰德尔逸度规则 Lewis-Randall rule of fugacity (54)露点 dew point (54)露点线 dew point line (54)麦克斯韦关系式 Maxwell relations (55)麦克斯韦速率分布 Maxwell distribution of speeds (55)麦克斯韦能量分布 MaxwelIdistribution of energy (55)毛细管凝结 condensation in capillary (55)毛细现象 capillary phenomena (55)米凯利斯常数 Michaelis constant (55)摩尔电导率 molar conductivity (56)摩尔反应焓 molar reaction enthalpy (56)摩尔混合熵 mole entropy of mixing (56)摩尔气体常数 molar gas constant (56)摩尔热容 molar heat capacity (56)摩尔溶解焓 mole dissolution enthalpy (56)摩尔稀释焓 mole dilution enthalpy (56)内扩散控制 internal diffusions control (56)内能 internal energy (56)内压力 internal pressure (56)能级 energy levels (56)能级分布 energy level distribution (57)能量均分原理 principle of the equipartition of energy (57)能斯特方程 Nernst equation (57)能斯特热定理 Nernst heat theorem (57)凝固点 freezing point (57)凝固点降低 lowering of freezing point (57)凝固点曲线 freezing point curve (58)凝胶 gelatin (58)凝聚态 condensed state (58)凝聚相 condensed phase (58)浓差超电势 concentration over-potential (58)浓差极化 concentration polarization (58)浓差电池 concentration cells (58)帕斯卡pascal (58)泡点 bubble point (58)泡点线 bubble point line (58)配分函数 partition function (58)配分函数的析因子性质 property that partition function to be expressed as a product of the separate partition functions for each kind of state (58)碰撞截面 collision cross section (59)碰撞数 the number of collisions (59)偏摩尔量 partial mole quantities (59)平衡常数（理想气体反应） equilibrium constants for reactions of ideal gases (59)平动配分函数 partition function of translation (59)平衡分布 equilibrium distribution (59)平衡态 equilibrium state (60)平衡态近似法 equilibrium state approximation (60)平衡状态图 equilibrium state diagram (60)平均活度 mean activity (60)平均活度系统 mean activity coefficient (60)平均摩尔热容 mean molar heat capacity (60)平均质量摩尔浓度 mean mass molarity (60)平均自由程mean free path (60)平行反应parallel reactions (61)破乳 demulsification (61)铺展 spreading (61)普遍化范德华方程 universal van der Waals equation (61)其它功 the other work (61)气化热heat of vaporization (61)气溶胶 aerosol (61)气体常数 gas constant (61)气体分子运动论 kinetic theory of gases (61)气体分子运动论的基本方程 foundamental equation of kinetic theory of gases (62)气溶胶 aerosol (62)气相线 vapor line (62)迁移数 transport number (62)潜热latent heat (62)强度量 intensive quantity (62)强度性质 intensive property (62)亲液溶胶 hydrophilic sol (62)氢电极 hydrogen electrodes (62)区域熔化zone melting (62)热 heat (62)热爆炸 heat explosion (62)热泵 heat pump (63)热功当量mechanical equivalent of heat (63)热函heat content (63)热化学thermochemistry (63)热化学方程thermochemical equation (63)热机 heat engine (63)热机效率 efficiency of heat engine (63)热力学 thermodynamics (63)热力学第二定律 the second law of thermodynamics (63)热力学第三定律 the third law of thermodynamics (63)热力学第一定律 the first law of thermodynamics (63)热力学基本方程 fundamental equation of thermodynamics (64)热力学几率 thermodynamic probability (64)热力学能 thermodynamic energy (64)热力学特性函数characteristic thermodynamic function (64)热力学温标thermodynamic scale of temperature (64)热力学温度thermodynamic temperature (64)热熵thermal entropy (64)热效应heat effect (64)熔点曲线 melting point curve (64)熔化热heat of fusion (64)溶胶 colloidal sol (65)溶解焓 dissolution enthalpy (65)溶液 solution (65)溶胀 swelling (65)乳化剂 emulsifier (65)乳状液 emulsion (65)润湿 wetting (65)润湿角 wetting angle (65)萨克尔－泰特洛德方程 Sackur-Tetrode equation (66)三相点 triple point (66)三相平衡线 triple-phase line (66)熵 entropy (66)熵判据 entropy criterion (66)熵增原理 principle of entropy increase (66)渗透压 osmotic pressure (66)渗析法 dialytic process (67)生成反应 formation reaction (67)升华热heat of sublimation (67)实际气体 real gas (67)舒尔采－哈迪规则 Schulze-Hardy rule (67)松驰力relaxation force (67)松驰时间time of relaxation (67)速度常数reaction rate constant (67)速率方程rate equations (67)速率控制步骤rate determining step (68)塔费尔公式 Tafel equation (68)态－态反应 state-state reactions (68)唐南平衡 Donnan equilibrium (68)淌度 mobility (68)特鲁顿规则 Trouton rule (68)特性粘度 intrinsic viscosity (68)体积功 volume work (68)统计权重 statistical weight (68)统计热力学 statistic thermodynamics (68)统计熵 statistic entropy (68)途径 path (68)途径函数 path function (69)外扩散控制 external diffusion control (69)完美晶体 perfect crystalline (69)完全气体 perfect gas (69)微观状态 microstate (69)微态 microstate (69)韦斯顿标准电池 Weston standard battery (69)维恩效应Wien effect (69)维里方程 virial equation (69)维里系数 virial coefficient (69)稳流过程 steady flow process (69)稳态近似法 stationary state approximation (69)无热溶液athermal solution (70)无限稀溶液 solutions in the limit of extreme dilution (70)物理化学 Physical Chemistry (70)物理吸附 physisorptions (70)吸附 adsorption (70)吸附等量线 adsorption isostere (70)吸附等温线 adsorption isotherm (70)吸附等压线 adsorption isobar (70)吸附剂 adsorbent (70)吸附量 extent of adsorption (70)吸附热 heat of adsorption (70)吸附质 adsorbate (70)析出电势 evolution or deposition potential (71)稀溶液的依数性 colligative properties of dilute solutions (71)稀释焓 dilution enthalpy (71)系统 system (71)系统点 system point (71)系统的环境 environment of system (71)相 phase (71)相变 phase change (71)相变焓 enthalpy of phase change (71)相变化 phase change (71)相变热 heat of phase change (71)相点 phase point (71)相对挥发度relative volatility (72)相对粘度 relative viscosity (72)相律 phase rule (72)相平衡热容heat capacity in phase equilibrium (72)相图 phase diagram (72)相倚子系统 system of dependent particles (72)悬浮液 suspension (72)循环过程 cyclic process (72)压力商 pressure quotient (72)压缩因子 compressibility factor (73)压缩因子图 diagram of compressibility factor (73)亚稳状态 metastable state (73)盐桥 salt bridge (73)盐析 salting out (73)阳极 anode (73)杨氏方程 Young’s equation (73)液体接界电势 liquid junction potential (73)液相线 liquid phase lines (73)一级反应first order reaction (73)一级相变first order phase change (74)依时计量学反应 time dependent stoichiometric reactions (74)逸度 fugacity (74)逸度系数 coefficient of fugacity (74)阴极 cathode (75)荧光 fluorescence (75)永动机 perpetual motion machine (75)永久气体 Permanent gas (75)有效能 available energy (75)原电池 primary cell (75)原盐效应 salt effect (75)增比粘度 specific viscosity (75)憎液溶胶 lyophobic sol (75)沾湿 adhesional wetting (75)沾湿功 the work of adhesional wetting (75)真溶液 true solution (76)真实电解质real electrolyte (76)真实气体 real gas (76)真实迁移数true transference number (76)振动配分函数 partition function of vibration (76)振动特征温度 characteristic temperature of vibration (76)蒸气压下降 depression of vapor pressure (76)正常沸点 normal point (76)正吸附 positive adsorption (76)支链反应 branched chain reactions (76)直链反应 straight chain reactions (77)指前因子 pre-exponential factor (77)质量作用定律mass action law (77)制冷系数coefficient of refrigeration (77)中和热heat of neutralization (77)轴功 shaft work (77)转动配分函数 partition function of rotation (77)转动特征温度 characteristic temperature of vibration (78)转化率 convert ratio (78)转化温度conversion temperature (78)状态 state (78)状态方程 state equation (78)状态分布 state distribution (78)状态函数 state function (78)准静态过程quasi-static process (78)准一级反应 pseudo first order reaction (78)自动催化作用 auto-catalysis (78)自由度 degree of freedom (78)自由度数 number of degree of freedom (79)自由焓free enthalpy (79)自由能free energy (79)自由膨胀free expansion (79)组分数 component number (79)最低恒沸点 lower azeotropic point (79)最高恒沸点 upper azeotropic point (79)最佳反应温度 optimal reaction temperature (79)最可几分布 most probable distribution (80)最可几速率 most propable speed (80)概念及术语BET公式BET formula1938年布鲁瑙尔(Brunauer)、埃米特(Emmett)和特勒(Teller)三人在兰格缪尔单分子层吸附理论的基础上提出多分子层吸附理论。

SPSS软件功能中英文对照Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOVA （analysis of variance）, 方差分析ANOVA Models, 方差分析模型Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs), BMDP统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interaction Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 一般线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常范围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样T ags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换。

T esting for Smooth Structural Changes in Time Series Models via Nonparametric RegressionBin ChenDepartment of EconomicsUniversity of RochesterandYongmiao HongDepartment of Economics and Statistical ScienceCornell University,Wang Yanan Institute for Studies in EconomicsXiamen UniversityJune,2008We would like to thank Mark Bils,Tim Bollerslev,Zongwu Cai,George H.Jakubson,Robert Jarrow,Nick Kiefer,Chung-ming Kuan,Oliver Linton,Joon Park,Jean-Francois Richard,Alan Stockman,George Tauchen and seminar participants at Cornell University,Duke University,National University of Singapore,University of Pittsburgh,University of Rochester and the2007Sino-Korean Econometrics Workshop in Xiamen,China for their comments.All remaining errors are solely ours.Hong acknowledges support from the National Science Foun-dation of China and the Cheung Kong Visiting Scholarship by the Chinese Ministry of Education and Xiamen University,China.Correspondences:Bin Chen,Department of Economics,University of Rochester,Rochester, NY14627;email:bchen8@;Yongmiao Hong,Department of Economics&Department of Sta-tistical Science,Cornell University,Ithaca,NY14850,USA,and Wang Yanan Institute for Studies in Economics (WISE),Xiamen University,Xiamen361005,China;email:yh20@.T esting for Smooth Structural Changes in Time Series Models viaNonparametric RegressionAbstractChecking parameter stability of economic models is a long-standing problem in time series econometrics.A classical econometric procedure is Chow’s(1960)test,which checks for the existence of a structural change on a known date.Various extensions have been made to test multiple changes with known or unknown change points.However,almost all existing structural change tests in econometrics are deigned to detect abrupt breaks.Little attention has been paid to smooth structural changes,which may be more realistic in economics.This paper proposes two consistent tests for smooth structural changes as well as abrupt structural breaks with known or unknown change points.The idea is to estimate the smooth time-varying parameters by local smoothing and compare them with the OLS parameter estimator.The…rst test compares the sums of squared residuals of the restricted constant parameter model and the unrestricted nonparametric time-varying parameter model,in a spirit similar to Chow’s(1960)F test.The second test compares the…tted values of the restricted and unrestricted models,which can be viewed as a generalization of Hausman’s(1978)test.Both tests have a convenient asymptotic N(0,1)distribution and do not require any prior information about the alternatives.Interestingly, unlike Chow’s(1960)test,the generalized Chow test is no longer optimal;it is asymptotically less powerful than the generalized Hausman test.A simulation study highlights the merits of the proposed tests in comparison with a variety of popular tests for structural changes.JEL Classi…cations:C1,C4,E0.Key words:Kernel,Chow’s test,Hausman’s test,Model stability,Nonparametric regression, Parameter constancy,Smooth structural change1.INTRODUCTIONDetection of structural changes in economic relationships is a long-standing problem in time series econometrics.It is particularly relevant for time series over a long time horizon since the underlying economic mechanism is likely to be disturbed by various factors.At the individual level,economic agents may change their production,consumption,savings and other patterns of behavior.This will lead to structural changes in an economic system,because econometric models consist of optimal decision rules of economic agents.A second driving force for structural changes are“shocks”induced by institutional changes,such as changes of government policies from a faith in monetarist policies to reliance on more Keynesian policies.Indeed,as Lucas (1976)points out,“any change in policy will systematically alter the structure of econometric models”,given that the structure of an econometric model depends crucially on agents’expecta-tions,which in turn vary systematically with changes in the structure of time series relevant to decision makers.Third,technological progress,which embraces the process of innovations and the long term determinants of capital accumulation,has dramatically changed economic funda-mentals and economic structures in the past decades.The prevalence of structural instability in macroeconomic time series relations has indeed been con…rmed by numerous empirical studies. For example,Stock and Watson(1996)test76representative U.S.monthly postwar macroeco-nomic time series and…nd substantial instability in a signi…cant fraction of the univariate and bivariate autoregressive models.Ben-David and Papell(1998)…nd statistically signi…cant evi-dence of a slowdown in the GDP growth in46countries.McConnell and Perez-Quiros(2000) and Kim,Nelson and Piger(2004)…nd overwhelming evidence of a substantial decrease in the volatility of the U.S.GDP growth rates in the early1980s.In investigating labor productivity in the U.S.manufacturing/durables sector,Hansen(2001)identi…es strong evidence of a structural break sometime between1992and1996,and weaker evidence of a structural break in the1960s and the early1980s.Econometric models usually specify a relationship between economic fundamentals.Econo-mists always hope that econometric models can capture important features of these relationships using a constant structure.Model stability is crucial for statistical inference,out-of-sample fore-casts,and any policy implications drawn from the model.Moreover,the existence of relatively constant linear relationships between economic variables is important for model parameters to be regarded as marginal propensities or elasticities.Such economic interpretations will be invalid in the presence of structural changes.Therefore,detection and identi…cation of structural changes are important and can be regarded as a…rst step to build a nonstationary time series model in view of modeling strategy.Not surprisingly,a large portion of the literature has been devoted to testing structural changes.See,e.g.,Hansen(2001),Banerjeea and Urga(2005)and Perron(2006)for excellentrecent surveys.A classical econometric procedure is Chow’s(1960)F test,which checks for the existence of an abrupt structural break at a known break date.This test has optimal power in the classical normal linear regression context for testing constancy of regression parameters across two or more regimes,each of which has enough data points to allow reliable estimation of the regression parameters,conditional on the error variances being constant across regimes. However,the F test becomes invalid even asymptotically when conditional heteroskedasticity exists,which has been documented as an empirical stylized fact in many economic and…nancial time series(e.g.,Engle1982,Bollerslev1986).Another important limitation of Chow’s(1960) test is that the break date must be known ex-ante,which is usually unattainable.1In practice, a break date may be decided on some known feature of the data.However,whether or not the obtained p-value takes into account the pre-test examination of the data can make a dramatic di¤erence:the signi…cance levels are usually much higher than the nominal ones that assume that the break point was chosen exogenously.There has been some research on testing structural breaks with unknown break dates.For example,Quandt(1960)proposes taking the largest Chow statistic over all possible break dates.If the break date is unknown a priori,however,the Chi-square critical values are inappropriate.This is a non-standard problem as the break date is only identi…ed under the alternative hypothesis.This problem is treated under various degrees of speci…city by Davies(1977),Hawkins(1987),Horvath(1995)and generalized by Andrews (1993).Andrews(1993)and Andrews and Ploberger(1994)consider a class of the supremum and the average of exponential Lagrange multiplier(LM),Wald,and likelihood ratio(LR) tests for structural breaks with unknown break dates.The test statistics have nonstandard asymptotic distributions and simulation methods are needed to obtain critical values.All these are speci…cally designed for the case of a single break although some have power against multiple breaks.The problem of multiple structural changes has received considerably less but increasing attention.Andrews,Lee and Ploberger(1999)develop the average and exponential F tests for the existence of changepoints at unknown times in a normal linear multiple regression.Bai and Perron(1998)promote the maximum of a sequence of F statistics for multiple breaks. The average,exponential and maximum F tests require the computation of the F test over all permissible partitions of the sample.Therefore the number to be evaluated is of the order O(T m) for a sample with T observations and m possible breaks.Moreover,Vogeslang(1999)concludes that most tests have nonmontonic power if the number of true breaks is greater than the number of breaks explicitly accounted for in the construction of the tests.For cases where the number of breaks is unknown,Bai and Perron(1998,2003)suggest a double maximum tests given some pre-1In rare cases,there are some natural canadidates for breakdates,such as the Black Monday,October19, 1987.speci…ed upper bound and Qu and Perron(2007)extend this analysis to a system of equations. However prior information on the upper bound may not be available and a large upper bound will lead to very involved computation.Another problem of the class of the supremum and average tests is that the statistics must be calculated in a trimmed range of the sample and hence power loss is inevitable.In addition to Chow’s(1960)test,many existing tests for structural changes in the literature are also based on the analysis of residuals.One which has played an important historical role is CUSUM test proposed by Brown,Durbin and Evans(1975)and further studied by Kramer, Ploberger and Alt(1988),Ploberger and Kramer(1992),among many others.They work on the maximum of partial sums of recursive or OLS residuals.A drawback of CUSUM test is that it exhibits only trivial power against alternatives in certain directions,as shown by Kramer et al.(1988)using asymptotic local power and by Garbade(1977)using simulations.Also,Hackl (1980)proposes MOSUM tests using the maximum of moving sums of recursive residuals and generalized by Kuan and Hornik(1995).Several additional tests in the literature such as tests of Leybourne and McCabe(1989),Nyblom(1989)and Hansen(1992)are designed for alternatives with stochastic trend and hence,have a di¤erent focus.Most existing tests for structural changes in time series econometrics are designed to detect abrupt structural breaks.Two exceptions are tests of Farley,Hinich and McGuire(1975)and Lin and Ter::a svirta(1994),who consider testing for parametric smooth structural changes.Smooth structural changes have received little attention in the literature.In fact,as Hansen(2001)points out,“it may seem unlikely that a structural break could be immediate and might seem more reasonable to allow a structural change to take a period of time to take e¤ect”.Technological progress,preference change and policy switch are some leading driving forces of structural changes that usually exhibit evolutionary changes in the long term.In economic modeling,parameters re‡ect economic agents’decision rules.Because it takes time for economic agents to collect and digest information needed for making decisions and for markets to reach some consensus due to heterogeneous beliefs,economic agents usually update their beliefs continuously given their information.On the other hand,even if individual agents could respond immediately to sudden changes,the aggregated economic variables over many individuals may become smooth. Aggregated data are measured by weighting the relative importance of heterogeneous subsets of economic units,where aggregating weights usually change smoothly with time.Moreover, di¤erent economic agents may switch their behaviors or expectations at di¤erent time.Hence,the aggregated economic variables will change smoothly rather than jump abruptly over time.In fact, evolutionary changes have been found as a rule rather than an exception in developing countries. For example,Sen(1989)shows that the poverty line,income gap ratio and Gini coe¢cient change smoothly since governments in developing countries impose direct or indirect taxes immediatelycorresponding to such extraneous shocks as an"oil embargo"and the lowering of oil prices. Government policies work as a bumper to absorb some e¤ects of sudden shocks that may cause abrupt structural breaks.Greenaway,Leybourne and Sapsford(1997)reappraise the time series properties of long-run economic growth rates in a number of developing countries which have undertaken liberalization.They conclude that even if the liberalization is implemented in a “big bang”manner,any subsequent e¤ects on growth will typically be gradual and the speed of adjustment depends on the e¢ciency of markets.During the past decade or so,time-varying time series models have appeared as a novel tool to model the evolutionary behavior of macroeconomic and…nancial time series.A leading example is the Smooth Transition Regression(STR)model,developed and popularized by Lin and Ter::a svirta(1994).Thanks to the use of a transition function,the STR model allows both the intercept and slope to change smoothly over time.Parametric models for time-varying parameters lead to more e¢cient estimation if the underlying coe¢cient functions are indeed correctly speci…ed.However,economic theories usually do not suggest any concrete functional form for time-varying parameters;the choice of a functional form is somewhat arbitrary.As mentioned by Phillips(2001),models for trending time series are rather complicated and use of nonparametric methods may be one solution.A nonparametric time-varying parameter time series model is…rst introduced by Robinson(1989,1991)and further studied by Orbe,Ferreira and Rodriguez-Poo(2000,2005,2006)and Cai(2007).One advantage of this nonparametric time-varying parameter model is that little restriction is imposed on the functional forms of both the time-varying intercept and slope,except for the condition that they evolve with time smoothly.Motivated by the‡exibility of the nonparametric time-varying parameter model,we will use it as the alternative to test smooth structural changes for a linear regression model.To our knowledge,there are only two tests designed explicitly for smooth structural changes in the literature.Farley et al.(1975)construct an F test by comparing a linear time series model with a parametric alternative whose slope is a linear function of time.Lin and Ter::a svirta(1994) develop LM type tests against a STR alternative,where the transition variable is a function of time.Simulations show that their tests have better power than CUSUM and‡uctuation tests when the data generating process(DGP)has continuous or non-monotonic structural changes. An undesired feature of these tests is that they use a speci…c parametric time-varying parameter model.While these tests have best power against the assumed alternative,no prior informa-tion about the true alternative is usually available for practitioners.In such scenarios,it is highly desirable to develop consistent tests that have good power against all-round alternatives of structural changes.This paper proposes two new consistent tests for smooth structural changes as well as abrupt structural breaks.Such tests complement the existing tests for abrupt structural breaks andavoid the di¢culty associated with whether there are multiple breaks and/or whether the break-dates are unknown.We estimate smooth-changing parameters by local linear regressions and compare them with the OLS parameter estimator.Local linear smoothing is a generalization of standard kernel smoothing.It includes the standard kernel smoothing as a special case and has the advantage of e¤ective bias-reducing near the boundary regions(e.g.,Fan and Yao2003). The proposed…rst test is a generalized Chow’s(1960)F test,based on the comparison of the sums of squared residuals between the restricted constant parameter regression model and the unrestricted nonparametric regression model.The second test can be viewed as a generalized Hausman’s(1978)test,based on the direct comparison between the OLS regression estimator and the nonparametric regression estimator.Interestingly,unlike Chow’s(1960)test,which is optimal in the context of the classical linear regression model with i.i.d.normal errors,the generalized Chow test is no longer optimal.It is shown that the generalized Hausman test is asymptotically more powerful than the generalized Chow test.Compared with the existing tests for structural breaks in the literature,the proposed tests have a number of appealing features.First,the proposed tests are consistent against a large class of smooth time-varying parameter alternatives.They are also consistent against multiple sudden structural breaks with unknown break points.Second,no prior information on a structural change alternative is needed.In particular,we do not need to know whether the structural changes are smooth or abrupt,and in the cases of abrupt structural breaks,we do not need to know the dates or the number of breaks.Third,unlike most tests for structural breaks in the literature,which often have nonstandard asymptotic distributions,the proposed tests have a convenient null asymptotic N(0,1)distribu-tion.The only inputs required are the OLS parameter estimator and the local linear time-varying parameter estimator.The latter is in fact a locally weighted least squares parameter estimator. Hence,any standard econometric software can carry out computational implementation easily.Fourth,the nonparametric time-varying parameter estimator is sensitive to the local behavior of time-varying parameters.Because only local information is employed in estimating parameters at each time point,the proposed tests have symmetric power against structural breaks that occur either in the…rst or second half of the sample period.This is di¤erent from some existing tests that have di¤erent powers against structural breaks that have same sizes but occur at di¤erent time points.Fifth,unlike some existing tests for structural breaks,no weighting function or trimming procedures are needed for the proposed tests.Moreover,as a by-product,the nonparametric local linear estimators of the potentially time-varying intercept and slope can provide insight into the economic relationship.In Section2,we introduce the framework and state the hypotheses of interest.Section3 describes the nonparametric approach and Section4derives the forms of test statistics and their asymptotic null distribution.Section5investigates the asymptotic power properties of the test statistics.In Section6,a simulation study is conducted to assess the reliability of the asymptotic theory in…nite samples.Section7provides concluding remarks and directions for future research.All mathematical proofs are collected in the appendix.A GAUSS code to implement the proposed tests is available from the author upon request.Throughout the paper, C denotes a generic bounded constant.2.HYPOTHESES OF INTERESTConsider the following DGPY t=X0t t+"t;t=1;:::;T;(2.1)where Y t is a dependent variable,X t is a d 1vector of explanatory variables, t is a d 1 possibly time-varying parameter vector,"t is an unobservable regression disturbance such that E("t j X t)=0almost surely(a.s.),d is a…xed positive integer,and T is the sample size.Theregressor vector X t can contain exogenous explanatory variables and lagged dependent variables. Thus,both static and dynamic regression models are covered.Like the bulk of the literature on structure changes,we are interested in testing constancy of the regression parameter in(2.1).The null hypothesis of interest isH0: t= for some constant vector and for all t:The alternative hypothesis H A is that H0is false.Under H0;the unknown constant parameter vector can be consistently estimated by the OLS estimator^ =arg min2R dTX t=1(Y t X0t )2:(2.2)Statistical inferences on the parameter can be made using the OLS estimator^ :Various inferential procedures have been available,depending on the assumption on the regression error "t(i.e.,whether f"t g is i.i.d.(0; 2);or whether there exists serial correlation and/or conditionalheteroskedasticity in f"t g).Detection and identi…cation of structural changes have been a long-standing problem in time series econometric modelling.Most existing works in the econometric literature are concernedwith abrupt structural breaks,either a single or multiple ones,and their locations.Little at-tention has been devoted to smooth structural changes.This is the main focus of the present paper.Under the alternative H A; t is a time-varying parameter.We are interested in the alternatives where t is a smooth deterministic function of t:A simple example of a smooth change parameter t is when X t=1:Then model(2.1)boils down to a deterministic trend model,which has been considered by Hall and Hart(1990),Johnstone and Silverman(1997),and Robinson(1997)for time domain smoothing.Because deterministic trends need not be polynomial(see Phillips2001), the nonparametric time trend model will allow more‡exibility in capturing the trend dynamics of Y t:2Another example of a smooth change parameter is a STR model,as developed by Granger and Ter::a svirta(1993)and Lin and Ter::a svirta(1994),wheret= 0+ 1F(t);0and 1are d 1parameter vectors,and F(t)is a smooth transition function.A choice of F(t)is a logistic function,namelyF(t)=f1+exp[ (t c)]g 1;where c and are scalar parameters governing the threshold and speed of the transition.By construction,tests for parametric smooth change alternatives have best power against the assumed alternative.Unfortunately,usually no prior information about the structural change alternative is available in practice.To cover a wide range of alternatives,we consider the following nonparametric time-varying parameter regression model:Y t=X0t t T +"t;t=1;:::;T;(2.3)where :[0;1]!R d is an unknown smooth function except for a…nite number of points over [0;1]:Discontinuities of ( )at a…nite number of points in[0;1]allow abrupt structural changes.This model is introduced by Robinson(1989,1991)and its nonparametric estimation has been considered in Robinson(1989,1991),Orbe et al.(2000,2005,2006)and Cai(2007).3It 2Juhl and Xiao(2005)propose a nonparametric test for structrual changes in a deterministic trend model, where the functional form of the trend is unknown.They…rst detrend the original data nonparametrically to avoid the bias in estimating parameters.Unlike existing tests in the literature,their test has monotonic power even when there are two changes in the trend.3These authors consider nonparametric estimation of time-varying parameters (t=T)and 2(t=T);where var("t)= 2(t=T):They do not consider testing parameter constancy.avoids restrictive parameterization of ( ).The speci…cation that parameter ( )is a function of ratio t=T rather than time t only is a common scaling scheme in the literature(see,e.g.,Phillips and Hansen1990,Robinson1991,Park and Hahn1999and Cai2007).It might…rst appear a bit strange because it makes the time-varying parameter t depend on the sample size T:The reason for this requirement is that a nonparametric estimator for t will not be consistent unless the amount of data on which it depends increases,and merely increasing the sample size will not necessarily improve estimation of t at some…xed point t;even if some smoothness condition is imposed on t:The amount of local information must increase suitably if the variance and bias of a nonparametric estimator of t are to decrease suitably.A convenient way to achieve this is to regard t as ordinates of smooth function ( )on an equally spaced grid over[0;1]; which becomes…ner as T!1;and consider estimation of ( )at…xed points :Consistent estimation of model(2.3)has been considered in Robinson(1991)and Cai(2007)using local constant smoothing and local linear smoothing respectively.The speci…cation of t= (t=T)does not regard the sampling of(Y t;X0t)0as taking place on a grid on[0;1];which would make the preservation of independence or weak dependence properties as T increases implausible.We note that the device of taking(Y t;X0t)0to be observations at intervals1=T on a continuous process on[0;1]that itself is independent of T would not work because it does not achieve the accumulation of new information as T increases,which is necessary for consistency.Making parameter t dependent on T is not unknown elsewhere in statistics and econometrics.4A well-known example is the local power analysis,where local alternatives are usually speci…ed as a function of the sample size T:Model(2.3)includes the class of locally stationary autoregressive models as considered in Dahlhaus(1996,1997):Y t= 0 t T +p X j=1 j t T Y t j+"t;where"t= (t=T)v t and v t i:i:d:N(0;1):Intuitively,locally stationary processes are nonsta-tionary time series whose behavior can be locally approximated by a stationary process.In time series econometrics,it is often assumed that nonstationary economic time series can be trans-formed,mainly by removing time trends and/or taking di¤erences,into a stationary process. In fact,the transformed time series may not be stationary,although trending behaviors have been removed.Locally stationary time series models nicely…ll this gap.They have attracted increasing attention in economics and…nance since they provide new insight into modelling such economic and…nancial variables as interest rates and stock prices(see,e.g.,Polasek1989, 4It may be noted that any asymptotic analysis is open to the criticism that it is impossible in many applications to increase T;the motivation being merely to provide approximate justi…cation for an inferential procedure based on a…nite sample.Stephan and Skander2004).We will assume that ( )is continuous except for a…nite number of points on[0;1]:In other words,we permit ( )to have…nitely many discontinuities.Hence,single structural break or multiple breaks with known or unknown break points,which are often considered in this literature,are special cases of model(2.3).For example,suppose ( )is a jump function,namely,( )=( 0;if 0;1;otherwise:Then we obtain the single break model considered in Chow(1960).3.NONPARAMETRIC TESTINGAlmost all existing tests for structural changes in the literature consider an alternative of abrupt structural changes.A di¢culty for testing H0lies in the fact that the possible change-points are usually unknown in practice so one needs to check for all t:Extending Chow’s(1960) test,which assumes a single break with a known break point,Andrews(1993),Andrews and Ploberger(1994),and Andrews et al.(1996)propose sup F T and exp F T tests,namely,sup F T sup 2 F T( );andexp F T=(1+c) mv=2X 2 exp mv2 c1+c F T( ) J( );where is the discrete set of the unspeci…ed change points,F T( )is the standard F test of structural change occurring at the time[T ];c and J( )are a constant and a weighting function respectively that have to be prespeci…ed by practitioners.Bai and Perron(1998,2003)further generalize Andrews’(1993)test to cases where the exact number of breaks is unknown but the upper bound is known.They develop two double maximum tests.The…rst is an equal-weight version,namely,UD max F T=max1 m Msup( 1;:::; m)2F T( 1; 2;:::; m);where F T( 1; 2;:::; m)is the F test of structural changes occurring at([T 1];[T 2];:::;[T m]) and M is a prespeci…ed upper bound of the number of breaks.The second test applies weights to the individual tests so that the marginal p values are equal across values of m;namely,W D max F T=max1 m M c( ;1)c( ;m)sup( 1; 2;:::; m)2F T( 1; 2;:::; m);where c( ;m)is the asymptotic critical value of the test sup(1; 2;:::; m)2F T( 1; 2;:::; m)for。

（转）Parameterestimationfortextanalysis暨LDA学习⼩结Reading Note : Parameter estimation for text analysis 暨LDA学习⼩结原⽂：伟⼤的Parameter estimation for text analysis！当把这篇看的差不多的时候，也就到了LDA基础知识终结的时刻了，意味着LDA基础模型的基本了解完成了。

所以对该模型的学习告⼀段落，下⼀阶段就是了解LDA⽆穷⽆尽的变种，不过那些不是很有⽤了，因为LDA已经被⼈⽔遍了各⼤“论坛”……抛开LDA背后复杂深⼊的数学背景不说，光就LDA的内容，确实不多，虽然变分法还是不懂，不过现在终于还是理解了“LDA is just a simple model”这句话。

总结⼀下学习过程：1.：CDF、PDF、Bayes’rule、各种简单的分布Bernoulli，binomial，multinomial、包括对prior、likelihood、postprior的理解（PRML1.2）2.共轭：？3.概率图模型 Probabilistic Graphical Models: PRML Chapter 8 基本概念即可4.采样算法：，Sampling Methods（PRML Chapter 11），，Gibbs Sampling5.6.进阶资料：《》、本⽂——————————————– 伟⼤的分割线！PETA！——————————————–⼀、前⾯⽆关部分⼆、模型进⼀步记忆从本图来看，需要记住：1.θm是每⼀个document单独⼀个θ，所以M个doc共有M个θm，整个θ是⼀个M*K的矩阵（M个doc，每个doc⼀个K维topic分布向量）。

2.φk总共只有K个，对于每⼀个topic，有⼀个φk，这些参数是独⽴于⽂档的，也就是对于整个corpus只sample⼀次。

不像θm那样每⼀个都对应⼀个⽂档，每个⽂档都不同，φk对于所有⽂档都相同，是⼀个K*V的矩阵（K个topic，每个topic⼀个V维从topic产⽣词的概率分布）。

A conceptual model of the cement hydration processR.KrstulovicÂ,P.DabicÂ*Faculty of Technology,University of Split,Teslina10/V,21000Split,CroatiaReceived17September1998;accepted3February2000AbstractCement hydration development has been analyzed on the basis of measured data obtained by the microcalorimetrical method.Based on the most appropriate model assumed,thermokinetic analysis has been carried out for the polymineral and polysize system tested on industrial cement samples.A computer program has been developed to determine specific kinetic parameters describing individual hydration processes. The program makes it possible to determine the controlling processes during hydration and their share.Certain kinetic parameters make it possible to observe differences in hydration of pure clinker minerals and of Portland cement in a subtler way.D2000Elsevier Science Ltd. All rights reserved.Keywords:Portland cement;Hydration;Kinetics;Modeling1.IntroductionCement hydration has been examined from different points of view,one of which is very often the kinetic analysis of the hydration process development,which can be analyzed by different methods[1±8].Bearing in mind the complexity of the hydration mechanism in polymineral and polysize systems,and the absence of a universal theoretical hypothesis in the study of heterogeneous sys-tems,it is necessary to take into account the methods of formal and conceptual mathematical modeling[9].This means that the mathematical analysis of data mea-sured during hydration is performed at several levels:data representation and forecast(the formal model),separation of hydration stages,and the thermokinetic analysis(the con-ceptual model).The first analysis of data measured is carried out in order to provide data necessary for the thermokinetic analysis(a±t data).The conceptual model is used to carry out the thermokinetic analysis of the hydration system in order to obtain parameters that finally describe hydration of pure clinker minerals,as well as the actual cement samples. Apart from the purely mathematical approach to modeling the hydration kinetics,subtle methods have been recently developed which make it possible to trace hydration of cement systems directly by means of sophisticated techni-ques that provide an image of the system,such as SEM and ESEM[10,11].A huge progress in description of hydration processes has been achieved by digital image analysis and by development of hydration models based on quantitative shares of individual minerals in the image[12],such as two-dimensional and three-dimensional NIST cement hydration model[13,14].In this way,it is possible to directly observe the development of hydrated particle microstructure,and to determine the reaction degree for individual constituents of the cement paste.This article outlines the relevant theoretical basis and the experimental thermokinetic analysis of the hydration process in the example of industrial cement according to the hypothe-tical mathematical model,which provides for the fact that the process takes place in a heterogeneous system according to characteristic laws of reactions taking place(nucleation and growth,interaction at phase boundaries,and diffusion).2.Theoretical principle for solving cement hydration kineticsThe kinetic model assumes three basic processes taking place:nucleation and crystal growth(NG),interactions at phase boundaries(I)and diffusion(D).All three processes are assumed to take place simultaneously but the slowest one dominates the hydration process as a whole,so that it is necessary to determine their rates.Based on studies carried*Corresponding author.Tel.:+385-21-385-633;fax:+385-21-384-964.E-mail address:dabic@ktf-split.hr(P.DabicÂ).0008-8846/00/$±see front matter D2000Elsevier Science Ltd.All rights reserved. PII:S0008-8846(00)00231-3Cement and Concrete Research30(2000)693±698out so far,it has been established that the slowest process atthe beginning is NG,which is later replaced by I or D,as can be concluded from the rate of these processes [15,16].The basic kinetic equations describing heterogeneous sys-tems are based on the change in the reaction degree relative to time elapsed (a ±t data).Microcalorimetry [17]is characterized by continuous determination of heat released during hydration of cement constituents,primarily clinker minerals,added gypsum,free CaO and MgO.By means of the heat released,it is possible to determine the degree of hydration relative to hydration duration according to the equation:a t Q t a Q max1where Q (t )is heat released by time t ,and Q max represents the total heat a sample can release.For pure minerals,a (t )is the mineral reaction degree,while for cement it is a conse-quence of the combined effect of heat released by all constituents present [18].2.1.Development of a mathematical model for kinetic analysis of cement hydrationThe basic Avrami±Erofeev equation describing hydra-tion kinetics during the dominant nucleation and growth process is often used for spherical particles: Àln 1Àa 13 K NG t2However,for a real system,the fixed exponent 1/3is usually replaced by 1/n where it is necessary to determine the most appropriate n .The value for the exponent n describes geometrical crystal growth [15]and its value usually ranges from 1to 3,and can be determined graphi-cally as a slope of the straight line from the log±log diagram of the relation Àln(1Àa )and elapsed time t .The method is illustrated by Fig.1.For interactions at phase boundaries,the following equa-tion is usually used:1À 1Àa 13 K I t3where K I is the rate constant for the reaction at phase boundaries,and is equal to k I /R .R represents the radius of the reacting particle.In microcalorimetry,the hydration degree a depends on the heat released as a consequenceof the total hydration effect,so that the effect of the change of radius of reacting particles during hydration is not examined.The diffusion process is defined by the expression: 1À 1Àa 13 2 K D t4where K D =k D /R 2and represents the diffusion process rate.The particle radius has not been considered further.Fig.1shows that a ±t data up to time t p are dominated by the NG process,i.e.it can be said for each a <a p and t <t p that they are dominated by the NG process.To determine the dominant process after t p ,the rates of these processes should be equated.If NG 3I or NG 3D,the following expressions are used:d a !NG d a !I or d a !NG d a!D 5 The basic Eqs.2±4should now be written in the differential form,so that they become Eqs.(6±8):d ad t !NG K NG n 1Àa Àln 1Àa n À1 n 6 d ad t !I 3K I 1Àa 23 7 d a d t !D 32K D 1Àa 23a 1À 1Àa 13 X 8 If expressions are equated according to Eq.5,the calculation constants K I0and K D0can be determined.These are control values to be compared to the experimental constants K I and K D .The condition is that the difference between the calculation and the experimental constants should be as small as possible.Value of the standard deviation,sd,represents the mean deviation of the experimental points in the interval chosen from the straight line drawn through these points,determined by the least square method.sd is calculated according to the Eq.(9):sd mi 1y i ÀY i 2N v u u u t9 where y i is the value of the experimental point,Y i represents the corresponding value on the straight line.N is the number of degrees of freedom on the interval observed,while the coefficients i and m represent the initial and final point of the interval observed.3.ExperimentalA referent alite (A)sample prepared in laboratory,and two types of commercial cement,a neat Portland cement,PC55designated as sample (B),and a blended Portland cement PC30t45s designated as sample (C)have beenusedFig.1.Graphical determination of the values for the exponent n from a ±t data,t p is the time up to which a ±t data are dominated by the NG process.R.KrstulovicÂ,P .Dabic Â/Cement and Concrete Research 30(2000)693±698694in this study.Cement samples B and C conform to the HRN B.C1-011standard.Sample B is actually cement with a compressive strength of 55MPa after 28days,while the C sample is blended Portland cement with 30%slag,with compressive strength of 45MPa after 28days.The miner-alogical composition of clinkers is shown in Table 1.Measurements of hydration heat have been carried out in a differential microcalorimeter for all samples under the fol-lowing conditions:mass sample =4g,W /C =0.5,T =25°C.4.Result analysisTo calculate the a ±t data,the signal formed due to the difference in potentials on the microcalorimeter thermocou-ples in the referent part and in the part with the hydrating sample,has been analysed by the PC software program [19]where the heat released Q (t )is calculated first,and then a (t )is determined according to the Eq.1.Fig.2shows the process of determining the transition time t p when hydration control is taken over by another process.Values K I and K I0show a high deviation,which means it is necessary to find the time for which a satisfactory correspondence is achieved.The precision of determination of the time interval I directly affects the values of the rate constant for the subsequent process,i.e.K D and K D0,as can be immediately seen on the graphic display at the computer screen,so that a compromise solution is searched for.The most appropriate transition time is determined by inputting a series of values for time in the vicinity of the assumed t p into the software program loop,and determining the most appropriate t p on the basis of correspondence between the theoretical con-stants and constants determined from the experimental data.Calculation of kinetic parameters based on the mathe-matical model starts with the determination of the time interval up to the point t p where the NG process is the dominant one,according to Fig.1.The control in choosing the optimum time for the t p point is the standard deviationTable 1Mineralogical composition of clinkers used to produce cements B and C calculated according to Bogue [R.H.Bogue,The chemistry of portland cement,2nd.ed.,Reinhold,New York (1955)]Sample C 3S C 2S C 3A C 4AF CaO free B 71.5022.70 3.02 1.75 1.00C62.1017.6011.208.200.90Fig.2.Diagram of determination of kinetic parameters from microcalorimetrical measurements:(1)calculated a ±t data,(2)kinetic analysis software,(3)differential microcalorimeter.R.KrstulovicÂ,P .Dabic Â/Cement and Concrete Research 30(2000)693±698695which shows how precisely the data line on the straight line is drawn through the experimental data by means of the least square method.Once the t p point has been determined,the program calculates the value of the exponent n and the rate constant K NG .Now,a menu makes it possible to test the remaining a ±t data after t p ,whereby determining the direction of the process according to data obtained for sd.In other words,selection is carried out for individual processes assumed and their domination is ascertained.The procedure can be seen in the flow diagram of the Basic ``Kinetic Analysis''program shown in Fig.3.The kinetic parameters determined for samples A,B,and C are shown in Table 2.Three measurements were made for the sample C,so that the marks C1,C2,and C3are used.5.DiscussionKinetic analysis of the early cement hydration stage is based on the mathematical model encompassing three basic dominant processes,NG,I,and D.The developed computer program [19]presents a determination of kineticparametersFig.3.A block diagram of the Q-BASIC program to determine kinetic parameters for the cement hydration process.*LSQÐa subroutine for the least squaremethod.It is also included in the calculation of processes I and D,which has not been shown here for greater clarity.Table 2Kinetic parameters determined by means of the mathematical hydration model ``Kinetic Analysis''NGIDSample n K NG (h À1)sd K I (m m h À1)sd K D (m m 2h À1)sd A 1.79600.03130.00040.01900.00140.00020.0002B 1.75100.05040.00580.01380.00150.00610.0014C1 1.92300.05290.00920.01320.00140.00410.0013C2 1.96800.05500.00770.01430.00130.00410.0066C31.92400.05610.00100.01300.00210.00320.0013R.KrstulovicÂ,P .Dabic Â/Cement and Concrete Research 30(2000)693±698696for cement hydration by means of values obtained for hydration heat on alite and Portland cement samples of different compositions.The hydration heat has been deter-mined by a differential microcalorimeter and the data have been gathered and processed up to48h.The computer software provided relevant kinetic data on all the parameters needed to define time intervals in which individual pro-cesses prevail.Values for kinetic parameters indicate that the first hours of hydration are dominated by the NG process.After that,the I process becomes the dominant process,followed by the D process.A comparison of kinetic parameters for the examined cement samples from Table2,with the clinker mineral alite as a referent sample,indicates essential differences in the systems examined.The values for the exponent n for alite and neat Portland cement(B)are relatively close,while the sample containing slag(C)has a much higher value of this exponent.This indicates that addition of slag to Portland cement greatly affects the crystal growth geometry.Values for K NG for cement samplesB andC are very similar,while this value for sample A differs to a great degree.This indicates that the polymineral composition of commercial cements greatly affects the nucleation and growth process. The standard deviation shows best agreement with the experimental data obtained for sample A,which could be expected,this being the pure clinker mineral.The values for the constants K I indicate similarity in samples B and C as compared to sample A.The diffusion process is most indicative with regard to the values of the parameter K D, the differences among individual samples being the largest there.A very low constant value up to48h characterizes alite,as well as a very small value of sd.For Portland cement samples,the constants are higher,but experimental data agree with the theoretical ones to a lesser degree as regards sd.Three measurements of the same cements samples marked C1,C2,and C3indicate satisfactory reproducibility of values for all kinetic parameters observed, as well as the share of individual processes up to48h hydration(Fig.4).Initial hydration time has not been covered by the mathematical analysis because of measuring problems caused by small temperature differences between the ce-ment paste and the water bath in the calorimeter[18].The contribution of heat developed due to wetting and dissol-ving cement particles in contact with water or to reactions of the free CaO,MgO,and gypsum with water has not been provided for either.If Refs.[15,20,21]are consid-ered,which had been obtained by mathematical analysis and the model for describing the process changes in hydration,which refer to individual clinker minerals, usually alite,a conclusion can be reached that the values in a real system such as cement show great differences relative to the composition and conditions for processing of data parisons of values for all kinetic parameters calculated according to the hydration model presented indicate that the data are within the boundaries obtained by different techniques for determination of the reaction degree[1,20].If parameters obtained for alite are compared to data according to Bezjak[15],where different exponents are considered for R(R2/3,R,R2)in the modified Eq.4,the closest values for the constant K D are for R2.This indicates that although the radius of reacting particles is not taken into consideration,the value of1/R2is included in the value of the relative rate constant K D,as has been assumed at the beginning K D=k D/R2.It can be said that the results obtained,shown in Table2, agree with the literature data if measuring conditions are taken into account.The comparison of measurements of different cement samples clearly indicates the difference in the share of individual processes(Fig.4).The graphical representation adds to the kinetic data in Table2with regard to changes in individual hydration processes with time relative to the cement type and some of its properties.Individual processes take place simultaneously,but affect hydration rate and particles of different sizes in a different manner.The model presented has been simplified;only the possibility of deter-mination of basic kinetic parameters by mathematical ana-lysis of the resultant plot for heat released during hydration of commercial cement has been considered.Flexibility of the model provides for further extensions by addition of mathematical modules for various factors affecting cement hydration,such as particle size and distribution,addition of admixtures,etc.A simple and relatively cheap technique that can be found in most cement production plants offers a perspective for research related to designing cement systems according to the requirements for the hydration process development based on a global experimental Q±t plot,i.e.a±t,which makes it possible to select the cement needed for specific purposes in a more exact way.6.ConclusionsThe method developed makes it possible to calculate kinetic parameters from microcalorimetricalmeasurements Fig.4.Share of individual processes up to48h of hydration.R.KrstulovicÂ,P.DabicÂ/Cement and Concrete Research30(2000)693±698697of cement hydration and to clearly differentiate the three basic hydration processes and separate them by their share in time.The thermokinetic analysis can provide certain guide-lines for further study of the hydration process development and changes in the values of kinetic parameters in indivi-dual periods.For samples B and C,the most probable process sequence is NG3I3D.Reproducibility of measurements has been confirmed,and satisfactory agreement of kinetic parameters (n,K NG,K I,and K D)has been obtained.The model used with cement samples in comparison to alite indicates that the results obtained are acceptable within certain limits.The kinetic analysis described and the software program developed provide for a rapid and easy approach to model-ing and determining individual development stages in early hydration periods and for cement samples,especially if factors affecting the early hydration are studied,which has been planned for supplementary research.AcknowledgmentsThe authors wish to express their gratitude to the Ministry of Science,Technology and Information of the Republic of Croatia,which has been financing a project,a part of which is presented in this report.References[1]S.Tsumura,Der hydratationsmechanismus der klinkerminerale in pas-teform,Zem-Kalk-Gips11(1966)511±518.[2]J.H.Taplin,On the hydration kinetics of hydraulic cements,V.Int.Symp.Chem.Cem.,Tokyo V ol.II,(1968)337±348.[3]K.Fujii,W.Kondo,Kinetics of the hydration of tricalcium silicate,JAm Ceram Soc57(1974)492±497.[4]C.Ostrowski,Z.Kowalczyk,Hydratationskinetik des zements,Baus-toffindustrie A4(1975)4±6.[5]A.Bezjak,Kinetic analysis of cement hydration including variousmechanistic concepts:I.Theoretical development,Cem Concr Res 13(1983)305±318.[6]K.-H.Schluessler,B.Kaessner,R.Sessner,M.Petke,Die hydrata-tionskinetik von zementÐmessung und mathematische modellierung, Silikattechnik40(1989)162±166.[7]herov-Marshak,O.P.Mchedlov-Petrossyan,A.G.Sinyakin,Non-isothermal thermokinetic of cement hardening,J Therm Anal 45(1995)985±991.[8]K.van Breugel,Numerical simulation of hydration and microstructur-al development in hardening cement-based materials,Heron37(1992) 1±62.[9]A.I.Korobov,A.M.Urshenko,herov-Marshak,Mathematicalmodels of thermokinetic analysis of hydration of binding materials, Cement11(1987)15±17.[10]Y.Wang,S.Diamond,An approach to quantitative image analysis forcement pastes,Mater Res Soc Symp Proc370(1995)23±33. [11]S.Mehta,R.Jones,B.Coveny,J.Chatterji,G.McPherson,Environ-mental Scanning Electron Microscope(ESEM)Examination of Indi-viduallly Hydrated Portland Cement Phases,16th International Conference on Cement Microscopy,Richmond(1994)239±257. [12]Y.Xi,P.D.Tennis,H.M.Jennings,Mathematical modelling of cementpaste microstructure by mosaic pattern:Part I.Formulation,J Mater Res11(1996)1943±1952.[13]D.P.Bentz,P.E.Stutzman,SEM analysis and computer modelling ofhydration of portland cement particles,in:S.Deflayes,D.Stark (Eds.),Petrografy of Cementitous Materials,ASTM STP1215, American Society for Testing and Materials,Philadelphia,1994, pp.60±73.[14]D.P.Bentz,Three-dimensional computer simulation of Portland ce-ment hydration and microstructure development,J Am Ceram Soc80 (1997)3±21.[15]A.Bezjak,Nuclei growth model in kinetic analysis of cement hydra-tion,Cem Concr Res16(1986)605±609.[16]A.Bezjak,I.JelenicÂ,On the determination of rate constants forhydration processes in cement pastes,Cem Concr Res10(1980) 553±563.[17]R.KrstulovicÂ,P.Krolo,T.FericÂ,Microcalorimetry in the cementhydration process,Periodica Polytechnica33(1989)315±321. [18]G.De Schutter,L.Taerwe,General hydration model for Portlandcement and blast furnace slag cement,Cem Concr Res25(1995) 593±604.[19]P.DabicÂ,Kinetic Analysis of Cement Hydration,InternationalCeramic Conference and ExibitionÐEUROFORUM'96,Ves-preÂm,Hungary.[20]K.van Breugel,Numerical simulation of hydration and microstructur-al development in hardening cement-based materials:II.Applications, Cem Concr Res25(1995)522±530.[21]A.Bezjak,I.JelenicÂ,V.Mlakar,A.PanovicÂ,A kinetic study of alitehydration,7th Int.Congress of the Chemistry of Cement,Paris V ol.II, (1980)111±116.R.KrstulovicÂ,P.DabicÂ/Cement and Concrete Research30(2000)693±698 698。

Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical modelsM.S.Ciupe a ,B.L.Bivort b ,D.M.Bortz c ,P.W.Nelsona,c,*aLos Alamos National Lab,MS K710,Los Alamos,NM 87545,United States b Department of Molecular and Cellular Biology,Harvard University,7Divinity Avenue,Cambridge,MA 02138,United States c Math Biology Research Group,Department of Mathematics,University of Michigan,530Church Street,Ann Arbor,MI 48109,United StatesReceived 24September 2003;received in revised form 12December 2005;accepted 18December 2005Available online 9February 2006AbstractThe dynamics of HIV-1infection consist of three distinct phases starting with primary infection,then latency and finally AIDS or drug therapy.In this paper we model the dynamics of primary infection and the beginning of latency.We show that allowing for time delays in the model better predicts viral load data when compared to models with no time delays.We also find that our model of primary infection pre-dicts the turnover rates for productively infected T cells and viral totals to be much longer than compared to data from patients receiving anti-viral drug therapy.Hence the dynamics of the infection can change dra-matically from one stage to the next.However,we also show that with the data available the results are highly sensitive to the chosen model.We compare the results using analysis and Monte Carlo techniques for three different models and show how each predicts rather dramatic differences between the fitted param-eters.We show,using a v 2test,that these differences between models are statistically significant and using a jackknifing method,we find the confidence intervals for the parameters.These differences in parameter esti-mations lead to widely varying conclusions about HIV pathogenesis.For instance,we find in our model with time delays the existence of a Hopf bifurcation that leads to sustained oscillations and that these 0025-5564/$-see front matter Ó2006Elsevier Inc.All rights reserved.doi:10.1016/j.mbs.2005.12.006*Corresponding author.Address:Department of Mathematics,University of Michigan,530Church Street,Ann Arbor,MI 48109,United States.Tel.:+1(734)7633408.E-mail address:pwn@ (P.W.Nelson)./locate/mbsMathematical Biosciences 200(2006)1–272M.S.Ciupe et al./Mathematical Biosciences200(2006)1–27oscillations could simulate the rapid turnover between viral strains and the appropriate CTL response necessary to control the virus,similar to that of a predator–prey type system.Ó2006Elsevier Inc.All rights reserved.Keywords:HIV;Primary infection;Time delays;Monte Carlo;Jackknifing;Model sensitivity1.IntroductionThe dynamics of HIV-1consist of three distinct phases starting with primary infection,then latency andfinally AIDS or drug therapy.Primary infection begins when the hostfirst becomes infected with HIV-1.During this period there is a rapid increase in the number of viral particles in the plasma which can reach well over10million copies per ml.While in primary infection the infectious HIV-1particles seek out target cells,mostly CD4+T cells,and infect them until a peak in viral concentration is reached.After which,due to limitations of target cells and/or the emergence of cytotoxic(or effector)T cells that target HIV-1infected cells,the viral load begins to decline.These two factors play a major role in the control of viral loads during primary infection.Primary infection ends when the viral loads reach a set point which defines the beginning of latency.Viral load at set point is a good indicator of the future of the disease.During latency, there can be numerous changes in the viral load totals but not to the extent seen in primary infec-tion.In many patients the behavior is characteristic of a damped oscillating system in which there are substantial oscillations in viral loads until a quasi-steady state level is reached.This state can last for many years until the virus becomes more active,i.e.,the patient develops AIDS,or the patient receives an anti-viral therapy which causes viral totals to greatly diminish.We examine the dynamics of primary infection and latency in this paper by considering a delay differential equation model,where the time delay represents the lag in effector cell activa-tion after initial infection.Adding the dynamics of the effector cells to previously existing mod-els increases the number of model parameters and so we employ a Monte Carlo numerical technique and jackknifing to estimate the parameters.We assume certain parameters to have the same value as given in Stafford et al.[1]while allowing for others,such as the growth and death rates of effector cells,the length of the time delay,and the death rates of produc-tively infected T cells and virus to be estimated.Our analyses give widely varying estimates of these kinetic rates when compared to the estimates in Stafford et al.[1],such as the death rates of productively infected T cells and virus,while providing very similar dynamicalfits to the patient data.These results show how sensitive thefits to patient data are to the chosen model,especially when the data is insufficient,i.e.,not enough data points for the studied period to distinguish between the models.Even though the parameter estimates are sensitive to the model,each model allows for a different interpretation of the biology,and we show through the analysis of each model its usefulness in future research of HIV primary infection. For instance,wefind for certain parameter ranges that the dynamics lead to sustained oscilla-tions in viral loads which implies the viral dynamics are actively changing during latency.These dynamics are not seen in the models without the time delays unless certain non-linearities are introduced[2].2.Basic modelThe models in this paper seek to describe the dynamics of HIV-1viral load during primary infection.Stafford et al.[1]proposed an ODE model that illustrates two possible immune re-sponses that may control the steady state level of virus after primary infection.Their model con-sisted of three types of cells:target cells(CD4+T cells),T,productively infected CD4+T cells,T*, and free virus,V.They assume target cells decay exponentially with no proliferation.Their model accounts for primary infection and latency differently then previous models[3–7]by including an exponential term to account for the cytotoxic activity of effector cells targeting infected cells.They showed that without this term their model was unable to account for the long term steady state viral loads seen in some of the patients.We pursued the idea of CD8+T cell reduction of produc-tively infected T cells by developing an equation for CTL’s,E.2.1.Effector cell equationDetermining the proper equation for effector cells is not straightforward due to the complexities of the biology.Previous work by DeBoer and Perelson[2]and Nowak and Bangham[8]consid-ered effector cell dynamics withd Ed t¼a E EIÀd E E;ð1Þwhere a E is the rate with which an effector cell,E,comes in contact with a productively infected T cell,I,and d E is the death rate of effector cells.Note that their notation assumes I instead of T*for these cells.However,a problem with this choice for effector cells is that it leads to a steady statecondition I ss¼d EE and thus the steady state dynamics of the productively infected T cells are inde-pendent of any viral parameters.Therefore,one needs to assume a steady state level of produc-tively infected T cells and set the parameters for effector cells based on this value.This then puts a constraint on the full model’s dynamics and thus limits its predictive ability.One could also use a more sophisticated equation for effector cells[2]but this would only intro-duce more variability into the model and thus take away from the focus of simply including the dynamics for effector cells.We found that if we considered(1)in our model that the Monte Carlo simulations did not always settle to a global minima,especially when wefixed c=3dayÀ1.Also, when wefixed the steady state levels as done in[2,8]wefind using(1)that the models’predicted maximal levels of effector cells during primary infection were on average less then0.124l lÀ1and thus well below the values from the literature.Data from the literature shows the average number of effector cells to range between375and1100l lÀ1[9,10].We chose to assume that the effector cells were activated at a rate proportional to the level of productively infected T cells.In our model,these cells become activated at a rate p and decay with a half life of1/d E.We will also consider a time delay in the activation term to account for the time needed by effector cells to recognize an infection.For our case we chose,d Ed t¼pTÃÀd E E.ð2ÞWe consider the CTL burst size to be dependent on the duration of antigen presentation.Exper-imental studies have shown that following a brief antigen encounter CD8+T cells divide into M.S.Ciupe et al./Mathematical Biosciences200(2006)1–273effector cells and the differentiation continues even after the antigenic stimulus has been removed. The capacity of CTLs to undergo prolonged division in the absence of antigen has been monitored in Listeria monocytogenes bacterial infection in vitro by Mercado et al.[11]and Wong and Pamer [12].van Stipdonk et al.[13]emphasized this idea by engineering antigen presenting cells which regulate CD8+T cells priming.The results in[13]were used by Antia et al.[14]to argue that the simple predator-prey type dynamics for CTL cell responses was not appropriate and therefore the level of effector cells is not simply proportional to the concentration of productively infected T cells.However,van Stipdonk et al.[15]followed up their paper in2001and showed that CD8+T cells that received brief antigenic stimulation undergo apoptosis much faster than those that receive sustained signals from antigen presenting cells.Hence,disproving their earlier claims and again leaving the question,of what is the best equation for effector cells,unanswered. Another issue to consider is the importance of costimulatory cytokines in the CTL proliferation pathway.Wong and Pamer et al.[12]used IL-2to stimulate CTL differentiation after antigen re-moval,and although CTLs are capable of producing their own IL-2[16],van Stipdonk has shown that a prolonged antigenic presence increases the capacity of CTLs to produce and respond to IL-2simulations[15].Finally,there are no studies that explain how the duration of antigenic stimulation regulates CTL responses in viral infections like HIV.One difference between the immune response in HIV infection and Listeria infection regards the number of CD8+T cells involved in the response.Listeria induces huge proliferation rates,while only1.6–18.4%of the total number of CD8+T cells respond to HIV infections[17].Also,the physiological relevance of this programmed immune response is limited by the fact that conclusions were determined by in vitro experiments.Hence,we are far from understanding the dynamics of CTL division cycles and further studies are needed to assess whether or not the immune response depends on the duration of antigenic exposure.2.2.Effector modelWe will consider both a non-delay model(effector model)and one with a time delay,tÀs,in the activation of the effector cells(Delay model1).In either case the steady state level of produc-tively infected T cells is now controlled by viral parameters.Wefind in our case that when we con-sider(2)in the model that the effector cells can range from1to980l lÀ1which is in the same range as found in[8]and also overlaps with the values given in[9,10].With this in mind we have the following set of equations as our basic model,d Td t¼sÀdTÀkVT;d TÃd t¼kVTÀd TÃÀd x ETÃ;d Vd t¼N d TÃÀcV;d E d t ¼pTÃÀd E E.ð3ÞThe initial conditions are T(0)=T0,T*(0)=0,V(0)=V0,E(0)=0.Here T,T*,V and E2Rþand all parameters are in Rþ.The constant s represents a source of healthy cells and d is their 4M.S.Ciupe et al./Mathematical Biosciences200(2006)1–27death rate.k is the infectivity rate,d is the death rate of the infected cells and d x represents the effectiveness of the immune response.N is the number of virus particles produced per infected cell and c is the viral clearance rate.The inclusion of the term d x ET*,allows for the removal of productively infected T cells due to a cell mediated immune response.Stafford et al.[1]provided for this response by using a compli-cated and seemingly arbitrary exponential function for effector cell killing where instead of the Àd T*Àd x ET*term they have d=d0+d1(V)where d1(V)=0for t<t1and d1(V)=f(t)V for t>t1and t1is the assumed start time of the CD8+response.They then definedfðtÞ¼b1þj eÀðtÀt1Þ=D T1Àb1þj eÀðtÀt1Þ=D T2ð4Þand comment that other forms for f can provide similar responses(Stafford model2).What we have done differently is allow for a general solution of E.2.3.Analysis of effector modelIn the absence of the virus,i.e.,non-infected steady state,the T cell population has a steady state value ofb T¼sdð5Þwith all other variables having a steady state value of zero.Eq.(3)has two steady state solutions, the non-infected steady state(b T;0;0;0)and the infected steady state(T;TÃ;V;E)whereTükN d d Ecd x pTÀd d Ed x p!T>cNk;V¼N dckN d d Ecd x pTÀd d Ed x p¼N dcTÃ;E¼pd EkN d d Ecd x pTÀd d Ed x p¼pd ETÃ;ð6Þwhere T isT¼kN d2d Ecd x pÀdþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikN d2d Ecd x pÀd2þ4s k2N2d2d Ec2d x pr2k2N2d2d Ec2d x pð7Þand is easily shown to be positive for all parameters.In order to study the stability of the steady states we linearize(3)about a given steady state, (denoted by a subscript‘ss’),to getA¼ÀdÀkV ss0ÀkT ss0kV ssÀdÀd x E ss kT ssÀd x TÃss0N dÀc00p0Àd EB BB@1C CC A.M.S.Ciupe et al./Mathematical Biosciences200(2006)1–275The characteristic equation for the linearized system isk4þa1k3þa2k2þa3kþa4¼0;ð8Þwherea1¼cþdþd x E ssþd EþdþkV ss;a2¼ðd E kþkcþk dþkd x E ssÞV ssþdd Eþd d Eþd dþdcþcd Eþd x E ss cþTÃss pdxþdd x E ssþd cÀd kT ss Nþd x E ss d E;a3¼ððkd x cþkd x d EÞV ssþdd x cþcd E d xþdd x d EÞE ssþðÀd kNd EÀd dkNÞT ssþðd d E kþd E kcþpkd x TÃss þd kcÞV ssþd cd Eþðpdd xþpd x cÞTÃssþd d E dþd dcþcd E d;a4¼ðkcd E d x E ssþd d E kcþTÃssd x cpkÞV ssþdcd E d x E ssþcd E d dÀd dkT ss Nd EþTÃss pdxcd.By the Routh–Hurwitz conditions,all eigenvalues of(8)have negative real part if and only if a1>0;a4>0;B1 a1a2Àa3>0and B1a3Àa1a4>0.ð9ÞProposition1.The non-infected steady state is stable if and only ifc Nk >sd.ð10ÞProof.The Jacobian matrix at the non-infected steady state isA¼Àd0Àksd0Àd ks00N dÀc00p0Àd EB BB@1C CC A;which produces the eigenvalues k1¼Àd;k2¼Àd E;k3;4¼ÀðcþdÞÆffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðcÀdÞ2þ4ksN dq2.ð11ÞHence it is easily seen that all eigenvalues are real and negative given(10)is satisfied.Under this assumption the non-infected steady-state is locally stable.hProposition2.The infected steady state(6)is stable if and only if the Routh–Hurwitz inequalities (9)evaluated at this steady state are satisfied.6M.S.Ciupe et al./Mathematical Biosciences200(2006)1–27Proof.From(6)we have that the infected steady-state exists if and only ifNkcT>1;which will make TÃ;V;E positive.Let y¼Nk TÀ1.We will show that a1,a4,a1a2Àa3are positive when y is positive.By substituting equalities(6)into the expressions of a1,a2,a3and a4we obtaina1¼y dþkN d2d E cd x pþcþdþdþd E;a2¼yk2N2d3d Ec2pdxTþkN d2d EpdxþkN d2d2Ecpdxþd d EþkN d dcTþkN d d EcTþdd Eþcd Eþcd;a3¼y2kN d3d2Ecpdxþyk2N2d3d EcpdxTþk2N2d3d2Ec2pdxTþkN d2d2Epdxþd d E cþd dd Eþcd E dþkN d dd EcT;a4¼y2kN d3d2Epdxþyk2N2d3d2EcpdxTþc d dd E.We now have a1,a2,a3and a4as polynomials in y with positive coefficients,so they are positive for positive values of y.The next step is to compute B1=a1a2Àa3which isB1¼y2kN d3d2ExþkN d3d Exþk2N2d4d2Ecp2dxþk2N2d4d3Ec2p2dxþk2N2d4d E2xTþk3N3d5d2Ec3p2dxTþd2d E!þy d cdþd dd Eþd d E cþd2d Eþd d2E þk2N2d3d2Ec2pdxTþk2N2d4d Ec2pdxTþ2k2N2d3d E dc2pdxTþkN d2dcTþkN d2d EcTþ2kN d2d2EpdxþkN d2d3EcpdxþkN d3d EpdxþkN d2d3EcpdxþckN d2d Epdxþ2kN d2d E dpdxþ2kN d2d2EdcpdxþTkN d d2EcþkN d2d EcþkN d d E dcþkN d2dcþkNd dþkNd E dþkN d d2cþcd2E þcd2þdd2Eþc2dþc2d Eþc d dþd2d Eþ2cd E dþd dd Eþd d E c.Here B1is a second degree polynomial in y with positive coefficients.Hence B1is always positive on the positive axis.The proof of the last relation B1a3Àa1a2>0is tedious and can be verified using maple to be true when(10)is violated.hWe will show later that the stability of the infected steady state can bifurcate into a periodic orbit when we include a time delay.M.S.Ciupe et al./Mathematical Biosciences200(2006)1–2773.Model with time delaysIn this section we introduce a time delay in the model(3)by assuming that the immune response at time t is generated by the infection of a cell T*at time tÀs,where s is constant.The model (Delay model1)then becomesd Td t¼sÀdTÀkVT;d TÃd t¼kVTÀd TÃÀd x ETÃ;d Vd t¼N d TÃÀcV;d E d t ¼pTÃðtÀsÞÀd E E;ð12Þwhere the term T*(tÀs)allows for a time delay between the moment of infection and the recog-nition of the infected cells by the cytotoxic CD8+T cells.The initial values are Tð0Þ¼T0;TÃðhÞ¼0;Vð0Þ¼V0;Eð0Þ¼0;h2½Às;0 .3.1.Analysis of Delay model1We have again two steady-states,the non-infected steady-state(b T;0;0;0)and the infected steady state(T;TÃ;V;E)where T;TÃ;V;E are given by(6).We want to see if the delay changes the stability of the steady states.The linearized system in vector form isd Yd t¼BYðtÞþCYðtÀsÞ;where B and C are defined byB¼ÀdÀkV ss0ÀkT ss0kV ssÀdÀd x E ss kT ssÀd x TÃss0N dÀc0000Àd EB BB@1C CC A;C¼0000000000000p00B BB@1C CC AandYðtÞ¼TðtÞTÃðtÞVðtÞEðtÞB BB@1C CC A.The characteristic equation for(12)is then given bydetðk I4ÀBÀeÀks CÞ¼0;8M.S.Ciupe et al./Mathematical Biosciences200(2006)1–27that is,HðkÞ¼k4þb1k3þb2k2þb3kþb4þb5k2eÀksþb6k eÀksþb7eÀks¼0;ð13Þwhereb1¼a1;b2¼a2Àpd x TÃss;b3¼a3ÀðcþkV ssþdÞpd x TÃss;b4¼a4ÀðkV ssþdÞpd x cTÃss;b5¼d x pTÃss;b6¼ðkV ssþdþcÞd x pTÃss;b7¼ðdþkV ssÞcd x pTÃss;where a1,a2,a3,a4are the coefficients of the characteristic polynomial for the effector model,i.e., s=0.It is known that the infected steady state is stable if all the eigenvalues given by(13)have neg-ative real parts.We know that(13)with the condition s=0has all roots in the left half plane if(9) is satisfied.By Rouche´’s Theorem the transcendental equation has roots with positive real parts, given it had negative roots for s=0,if and only if it has purely imaginary roots.Our goal is to find under what conditions we will have no such roots.Recent results have shown that this calculation can be made by applying sturm sequences(see[18]).Let k=l(s)+i m(s)be the root of(13),with l and m2R.If we substitute into(13)we getHðl;mÞ¼½l4þm4À6l2m2þb1ðl3À3lm2Þþb2ðl2Àm2Þþb3lþb4þb5eÀlsððl2Àm2ÞcosðmsÞþ2lm sinðmsÞÞþb6eÀlsðl cosðmsÞþm sinðmsÞÞþb7eÀls cosðmsÞþi½4l3mÀ4lm3þb1ð3l2mÀm3Þþ2b2lmþb3mþb5eÀlsð2lm cosðmsÞÀðl2Àm2ÞsinðmsÞÞþb6eÀlsðm cosðmsÞÀl sinðmsÞÞÀb7eÀls sinðmsÞ .If we canfind a root of(13)of the form k=i m,then the infected steady state losses its stability. This will happen if and only ifHð0;mÞ¼½m4Àb2m2þb4Àb5m2cosðmsÞþb6m sinðmsÞþb7cosðmsÞþi½Àb1m3þb3mþb5m2sinðmsÞþb6m cosðmsÞÀb7sinðmsÞ ¼0ð14Þfor some m2R.It is clear that we have a solution m of(14)if both the real and imaginary parts of the equation equal zero.In other words,m4Àb2m2þb4¼b5m2cosðmsÞÀb6m sinðmsÞÀb7cosðmsÞÀb1m3þb3m¼Àb5m2sinðmsÞÀb6m cosðmsÞþb7sinðmsÞ.If we add the square of both equations we obtainm8þam6þbm4þcm2þx¼0;ð15Þwhere a¼b21À2b2,b¼b22þ2b4À2b1b3Àb25,c¼À2b2b4þb23Àb26þ2b5b7,and x¼b24Àb27.Ifwe substitute w=m2into(15)we obtain a fourth degree polynomial in wM.S.Ciupe et al./Mathematical Biosciences200(2006)1–279w4þa w3þb w2þc wþx¼0.ð16ÞIf all the roots of the polynomial(16)are negative or imaginary then(15)will have no roots since m is real.Hence,we have no purely imaginary roots for H,and the infected steady state will be stable in the delay case.If we reconsider(16),a fourth degree polynomial with real coefficients,we can apply the Routh–Hurwitz conditions to show when all the roots lie in the left half plane.Hence,we have the following result.Proposition3.Under the following assumptions:a>0;x>0;abÀc>0;abcÀaxÀc2>0.Eq.(16)has no positive real roots.4.Logistic model for target cellsIn our Delay model1(12)we have assumed a constant source term for target cells and an expo-nential death rate,however,it is possible to consider other behaviors such as logistic growth[2]. Assuming target cells can grow at a rate,a dayÀ1and that this growth is limited by a carrying capacity,T max cells,we haved T d t ¼a T1ÀTþTÃT maxÀkVT;d TÃd t¼kVTÀd TÃÀd x ETÃ;d Vd t¼N d TÃÀcV;d E d t ¼pTÃÀd E Eð17Þwith initial conditions T(0)=T0,T*(0)=0,V(0)=V0,E(0)=0.T max is the total number of target cells in the plasma.4.1.Analysis of logistic modelIn the absence of the virus the T-cell population has a steady state value of b T¼T max.Now if we consider an infection modeled by the system(17)we will have two steady state solutions,the non-infected steady stateðb T;0;0;0Þand the infected steady state(b T;b TÃ;b V;b E)whereb TükN d d Ecd x p b TÀd d Ed x p;b V¼N dckN d d Ecd x pb TÀd d Ed x p;10M.S.Ciupe et al./Mathematical Biosciences200(2006)1–27b E¼pd EkN d d Ecd x pb TÀd d Ed x p;b T¼aþkN d2d Ecd x pþa d E dd x pTmaxaT maxþa N d kd Ecd x pTmaxþk N2d d Ec2d x p.We linearize our system(where the subscript‘ss’is referring to any given steady state)to study stability andfind the Jacobian matrix to beA¼aÀa TÃssT maxÀ2a T ssT maxÀkV ssÀa T ssT maxÀkT ss0kV ssÀdÀd x E ss kT ssÀd x TÃss0N dÀc00p0Àd EB BB BB@1C CC CC A.ð18ÞThe corresponding characteristic equation for the linearized system isk4þ^a1k3þ^a2k2þ^a3kþ^a4¼0;ð19Þwhere^a1¼Àaþd EþdþcþkV ssþd x E ssþa TÃssT maxþ2a T ssT max;^a2¼a d xT maxTÃssþkd x V ssþ2a d xT maxT ssþd xðcþd EÀaÞE ssþða cþadþT max pd xþa d EÞT maxTÃssþk aT maxT ssþkcþkd Eþk dV ssþ2aðcþd EþdÞT maxÀN d kT ssþðÀadþd d Eþcd EÀa d Eþd cÀa cÞ;^a3¼ðkd x cþkd x d EÞV ssþðd x cd EÀa d x cÀa d x d EÞþa d x cþa d x d ET maxTÃssþ2a d x cþ2a d x d ET maxT ssE ssþðkd E aþk a cÞT maxT ssþkpd x TÃssþðk d cþkcd Eþk d d EÞV ssþðÀad d Eþd cd EÀa cd EÀad cÞþa pdxT maxTÃss2þð2a pd xÀk a N dÞT maxT ssþða cd Eþad cþT max pd x cÀa T max pd xþad d EÞT maxTÃssþa T max N d kþ2a cd Eþ2ad d EÀT max N d kd Eþ2ad cT maxT ssÀ2a N d kT maxT2ss;^a4¼a cd x d ET maxTÃssþkd x cd E V ssÀa cd x d Eþ2a d x cd E T ssT maxE ssþkpd x cTÃss þk d cd Eþkcd E a T ssT maxV ssþa N d kd E T ssÀad cd Eþ2ad cd ET maxT ssÀ2a N d kd ET maxT2ssþa pdxcT maxTÃss2þad cd Eþ2a cpd x T ssÀad Nkd E T ssÀa pcd x T maxT maxTÃss.。

几何布朗运动参数估计几何布朗运动是一种随机游走模型，它描述了一个粒子在空间中按照随机步长和随机方向运动的轨迹。

这种运动最早由数学家飞利浦·布朗于1827年观察到，被称为布朗运动。

几何布朗运动是布朗运动的一种变形，其步长由随机变量定义，并且可以在每一步上进行缩放。

估计几何布朗运动的参数是指基于已观测到的数据来计算运动模型中的未知参数。

这些参数包括步长的均值、方差，以及可能的相关性和非线性性质。

之后，这些参数可以用于模拟和预测未来的运动轨迹。

参数估计可以通过不同的方法来实现。

以下是一些常见的参数估计方法：1. 极大似然估计（Maximum Likelihood Estimation, MLE）：极大似然估计是一种常见的参数估计方法，它通过最大化样本数据出现的概率来估计参数值。

对于几何布朗运动，可以通过最大化观测到的步长数据的似然函数来估计参数。

2. 最小二乘估计（Least Squares Estimation, LSE）：最小二乘估计是一种通过最小化观测数据与模型预测值的差异来估计参数值的方法。

对于几何布朗运动，可以通过最小化步长数据与模型预测步长之间的差异来估计参数。

3. 粒子滤波（Particle Filter）：粒子滤波是一种基于蒙特卡洛模拟的参数估计方法，它通过将参数值表示为一组随机样本（粒子），在每个时刻根据观测数据的信息更新粒子的权重，并利用权重进行估计。

4. 贝叶斯统计（Bayesian Statistics）：贝叶斯统计是一种基于贝叶斯定理的概率方法，通过将参数值看作是未知的随机变量，并利用观测数据和先验知识来计算参数的后验分布。

通过对参数的采样可以进行参数估计。

除了上述方法，还有其他一些参数估计方法，如卡尔曼滤波和马尔可夫链蒙特卡洛方法等。

不同的估计方法适用于不同的数据和模型假设，选择合适的估计方法对于准确估计参数值非常重要。

综上所述，针对几何布朗运动的参数估计可以通过多种方法，如极大似然估计、最小二乘估计、粒子滤波和贝叶斯统计等来实现。